K. B. Nakshatrala

M. K. Mudunuru, K. B. Nakshatrala

This paper concerns with mesh restrictions that are needed to satisfy several important mathematical properties -- maximum principles, comparison principles, and the non-negative constraint -- for a general linear second-order elliptic partial differential equation. We critically review some recent developments in the field of discrete maximum principles, derive new results, and discuss some possible future research directions in this area. In particular, we derive restrictions for a three-node triangular (T3) element and a four-node quadrilateral (Q4) element to satisfy comparison principles, maximum principles, and the non-negative constraint under the standard single-field Galerkin formulation. Analysis is restricted to uniformly elliptic linear differential operators in divergence form with Dirichlet boundary conditions specified on the entire boundary of the domain. Various versions of maximum principles and comparison principles are discussed in both continuous and discrete settings. In the literature, it is well-known that an acute-angled triangle is sufficient to satisfy the discrete weak maximum principle for pure isotropic diffusion. An iterative algorithm is developed to construct simplicial meshes that preserves discrete maximum principles using existing open source mesh generators. Various numerical examples based on different types of triangulations are presented to show the pros and cons of placing restrictions on a computational mesh. We also quantify local and global mass conservation errors using representative numerical examples, and illustrate the performance of metric-based meshes with respect to mass conservation.

M. K. Mudunuru, K. B. Nakshatrala

This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non-linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non-negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always non-negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice-versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response.

K. B. Nakshatrala, A. J. Valocchi

We consider the tensorial diffusion equation, and address the discrete maximum-minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum-minimum principle) of mixed finite element formulations. The discrete maximum-minimum principle is the discrete version of the maximum-minimum principle. In this paper we present two non-negative mixed finite element formulations for tensorial diffusion equations based on constrained optimization techniques (in particular, quadratic programming). These proposed mixed formulations produce non-negative numerical solutions on arbitrary meshes for low-order (i.e., linear, bilinear and trilinear) finite elements. The first formulation is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative constraint to the variational statement of the Raviart-Thomas formulation. The second non-negative formulation based on the variational multiscale formulation. For the former formulation we comment on the affect of adding the non-negative constraint on the local mass balance property of the Raviart-Thomas formulation. We also study the performance of the active set strategy for solving the resulting constrained optimization problems. The overall performance of the proposed formulation is illustrated on three canonical test problems.

G. S. Payette, K. B. Nakshatrala, J. N. Reddy

The main aim of this paper is to document the performance of $p$-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steady-state) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation). We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used $p = 1, ..., 10$. It will be shown that the violation of the non-negative constraint will not vanish with $p$-refinement for anisotropic diffusion. We shall illustrate the performance of $p$-refinement using several representative problems. The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.

K. B. Nakshatrala, M. K. Mudunuru, A. J. Valocchi

We present a novel computational framework for diffusive-reactive systems that satisfies the non-negative constraint and maximum principles on general computational grids. The governing equations for the concentration of reactants and product are written in terms of tensorial diffusion-reaction equations. % We restrict our studies to fast irreversible bimolecular reactions. If one assumes that the reaction is diffusion-limited and all chemical species have the same diffusion coefficient, one can employ a linear transformation to rewrite the governing equations in terms of invariants, which are unaffected by the reaction. This results in two uncoupled tensorial diffusion equations in terms of these invariants, which are solved using a novel non-negative solver for tensorial diffusion-type equations. The concentrations of the reactants and the product are then calculated from invariants using algebraic manipulations. The novel aspect of the proposed computational framework is that it will always produce physically meaningful non-negative values for the concentrations of all chemical species. Several representative numerical examples are presented to illustrate the robustness, convergence, and the numerical performance of the proposed computational framework. We will also compare the proposed framework with other popular formulations. In particular, we will show that the Galerkin formulation (which is the standard single-field formulation) does not produce reliable solutions, and the reason can be attributed to the fact that the single-field formulation does not guarantee non-negative solutions. We will also show that the clipping procedure (which produces non-negative solutions but is considered as a variational crime) does not give accurate results when compared with the proposed computational framework.

K. B. Nakshatrala, S. H. S. Joodat, R. Ballarini

Geo-materials such as vuggy carbonates are known to exhibit multiple spatial scales. A common manifestation of spatial scales is the presence of (at least) two different scales of pores, which is commonly referred to as double porosity. To complicate things, the pore-network at each scale exhibits different permeability, and these networks are connected through fissure and conduits. Although some models are available in the literature, they lack a strong theoretical basis. This paper aims to fill this lacuna by providing the much needed theoretical foundations of the flow in porous media which exhibit double porosity/permeability. We first obtain a mathematical model for double porosity/permeability using the maximization of rate of dissipation hypothesis, and thereby providing a firm thermodynamic underpinning. We then present, along with mathematical proofs, several important mathematical properties that the solutions to the double porosity/permeability model satisfy. These properties are important in their own right as well as serve as good (mechanics-based) a posteriori measures to assess the accuracy of numerical solutions. We also present several canonical problems and obtain the corresponding analytical solutions, which are used to gain insights into the velocity and pressure profiles, and the mass transfer across the two pore-networks. In particular, we highlight how the solutions under the double porosity/permeability differ from the corresponding solutions under Darcy equations.

S. H. S. Joodat, K. B. Nakshatrala, R. Ballarini

The flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon-dioxide sequestration. The flow within numerous natural and synthetic porous materials that contain multiple scales of pores cannot be adequately described by the classical Darcy equations. It is for this reason that mathematical models for fluid flow in media with multiple scales of pores have been proposed in the literature. However, these models are analytically intractable for realistic problems. In this paper, a stabilized mixed four-field finite element formulation is presented to study the flow of an incompressible fluid in porous media exhibiting double porosity/permeability. The stabilization terms and the stabilization parameters are derived in a mathematically and thermodynamically consistent manner, and the computationally convenient equal-order interpolation of all the field variables is shown to be stable. A systematic error analysis is performed on the resulting stabilized weak formulation. Representative problems, patch tests and numerical convergence analyses are performed to illustrate the performance and convergence behavior of the proposed mixed formulation in the discrete setting. The accuracy of numerical solutions is assessed using the mathematical properties satisfied by the solutions of this double porosity/permeability model. Moreover, it is shown that the proposed framework can perform well under transient conditions and that it can capture well-known instabilities such as viscous fingering.

J. Chang, K. B. Nakshatrala, M. G. Knepley et al.

Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to determine whether a computational framework is performing efficiently or is scalable. The aim of this paper is to present various strategies for better understanding the performance of any parallel computational frameworks for solving PDEs. Important performance issues that negatively impact time-to-solution are discussed, and we propose a performance spectrum analysis that can enhance one's understanding of critical aforementioned performance issues. As proof of concept, we examine commonly used finite element simulation packages and software and apply the performance spectrum to quickly analyze the performance and scalability across various hardware platforms, software implementations, and numerical discretizations. It is shown that the proposed performance spectrum is a versatile performance model that is not only extendable to more complex PDEs such as hydrostatic ice sheet flow equations, but also useful for understanding hardware performance in a massively parallel computing environment. Potential applications and future extensions of this work are also discussed.

J. Chang, S. Karra, K. B. Nakshatrala

It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.

A. Rezaei, K. B. Nakshatrala, F. Siddiqui et al.

We present a systematic global sensitivity analysis using the Sobol method which can be utilized to rank the variables that affect two quantity of interests -- pore pressure depletion and stress change -- around a hydraulically-fractured horizontal well based on their degree of importance. These variables include rock properties and stimulation design variables. A fully-coupled poroelastic hydraulic fracture model is used to account for pore pressure and stress changes due to production. To ease the computational cost of a simulator, we also provide reduced order models (ROMs), which can be used to replace the complex numerical model with a rather simple analytical model, for calculating the pore pressure and stresses at different locations around hydraulic fractures. The main findings of this research are: (i) mobility, production pressure, and fracture half-length are the main contributors to the changes in the quantities of interest. The percentage of the contribution of each parameter depends on the location with respect to pre-existing hydraulic fractures and the quantity of interest. (ii) As the time progresses, the effect of mobility decreases and the effect of production pressure increases. (iii) These two variables are also dominant for horizontal stresses at large distances from hydraulic fractures. (iv) At zones close to hydraulic fracture tips or inside the spacing area, other parameters such as fracture spacing and half-length are the dominant factors that affect the minimum horizontal stress. The results of this study will provide useful guidelines for the stimulation design of legacy wells and secondary operations such as refracturing and infill drilling.

D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad

In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions.

K. B. Nakshatrala, A. Prakash, K. D. Hjelmstad

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy'' method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).

H. Nagarajan, K. B. Nakshatrala

In this paper, we consider anisotropic diffusion with decay, and the diffusivity coefficient to be a second-order symmetric and positive definite tensor. It is well-known that this particular equation is a second-order elliptic equation, and satisfies a maximum principle under certain regularity assumptions. However, the finite element implementation of the classical Galerkin formulation for both anisotropic and isotropic diffusion with decay does not respect the maximum principle. We first show that the numerical accuracy of the classical Galerkin formulation deteriorates dramatically with increase in the decay coefficient for isotropic medium and violates the discrete maximum principle. However, in the case of isotropic medium, the extent of violation decreases with mesh refinement. We then show that, in the case of anisotropic medium, the classical Galerkin formulation for anisotropic diffusion with decay violates the discrete maximum principle even at lower values of decay coefficient and does not vanish with mesh refinement. We then present a methodology for enforcing maximum principles under the classical Galerkin formulation for anisotropic diffusion with decay on general computational grids using optimization techniques. Representative numerical results (which take into account anisotropy and heterogeneity) are presented to illustrate the performance of the proposed formulation.

M. S. Joshaghani, J. Chang, K. B. Nakshatrala et al.

The objective of this paper is twofold. First, we propose two composable block solver methodologies to solve the discrete systems that arise from finite element discretizations of the double porosity/permeability (DPP) model. The DPP model, which is a four-field mathematical model, describes the flow of a single-phase incompressible fluid in a porous medium with two distinct pore-networks and with a possibility of mass transfer between them. Using the composable solvers feature available in PETSc and the finite element libraries available under the Firedrake Project, we illustrate two different ways by which one can effectively precondition these large systems of equations. Second, we employ the recently developed performance model called the Time-Accuracy-Size (TAS) spectrum to demonstrate that the proposed composable block solvers are scalable in both the parallel and algorithmic sense. Moreover, we utilize this spectrum analysis to compare the performance of three different finite element discretizations (classical mixed formulation with H(div) elements, stabilized continuous Galerkin mixed formulation, and stabilized discontinuous Galerkin mixed formulation) for the DPP model. Our performance spectrum analysis demonstrates that the composable block solvers are fine choices for any of these three finite element discretizations. Sample computer codes are provided to illustrate how one can easily implement the proposed block solver methodologies through PETSc command line options.

K. B. Nakshatrala, H. Nagarajan, M. Shabouei

Transient diffusion equations arise in many branches of engineering and applied sciences (e.g., heat transfer and mass transfer), and are parabolic partial differential equations. It is well-known that, under certain assumptions on the input data, these equations satisfy important mathematical properties like maximum principles and the non-negative constraint, which have implications in mathematical modeling. However, existing numerical formulations for these types of equations do not, in general, satisfy maximum principles and the non-negative constraint. In this paper, we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion equation. The method of horizontal lines (also known as the Rothe method) is applied in which the time is discretized first. This results in solving steady anisotropic diffusion equation with decay equation at every discrete time level. The proposed methodology for transient anisotropic diffusion equation will satisfy maximum principles and the non-negative constraint on general computational grids, and with no additional restrictions on the time step. We illustrate the performance and accuracy of the proposed formulation using representative numerical examples. We also perform numerical convergence of the proposed methodology. For comparison, we also present the results from the standard single-field semi-discrete formulation and the results from a popular software package, which all will violate maximum principles and the non-negative constraint.

J. Chang, K. B. Nakshatrala

Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection-diffusion equations. An attractive feature of the proposed framework is that it works with with any weak formulation for the advection-diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework.

K. B. Nakshatrala, M. S. Joshaghani

Many processes in nature (e.g., physical and biogeochemical processes in hyporheic zones, and arterial mass transport) occur near the interface of free-porous media. A firm understanding of these processes needs an accurate prescription of flow dynamics near the interface which (in turn) hinges on an appropriate description of interface conditions along the interface of free-porous media. Although the conditions for the flow dynamics at the interface of free-porous media have received considerable attention, many of these studies were empirical and lacked a firm theoretical underpinning. In this paper, we derive a complete and self-consistent set of conditions for flow dynamics at the interface of free-porous media. We first propose a principle of virtual power by incorporating the virtual power expended at the interface of free-porous media. Then by appealing to the calculus of variations, we obtain a complete set of interface conditions for flows in coupled free-porous media. A noteworthy feature of our approach is that the derived interface conditions apply to a wide variety of porous media models. We also show that the two most popular interface conditions -- the Beavers-Joseph condition and the Beavers-Joseph-Saffman condition -- are special cases of the approach presented in this paper. The proposed principle of virtual power also provides a minimum power theorem for a class of flows in coupled free-porous media, which has a similar mathematical structure as the ones enjoyed by flows in uncoupled free and porous media.

D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad

In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.

N. V. Jagtap, M. K. Mudunuru, K. B. Nakshatrala

Emerging technologies like hypersonic aircraft, space exploration vehicles, and batteries avail fluid circulation in embedded microvasculatures for efficient thermal regulation. Modeling is vital during these engineered systems' design and operational phases. However, many challenges exist in developing a modeling framework. What is lacking is an accurate framework that (i) captures sharp jumps in the thermal flux across complex vasculature layouts, (ii) deals with oblique derivatives (involving tangential and normal components), (iii) handles nonlinearity because of radiative heat transfer, (iv) provides a high-speed forecast for real-time monitoring, and (v) facilitates robust inverse modeling. This paper addresses these challenges by availing the power of physics-informed neural networks (PINNs). We develop a fast, reliable, and accurate Scientific Machine Learning (SciML) framework for vascular-based thermal regulation -- called CoolPINNs: a PINNs-based modeling framework for active cooling. The proposed mesh-less framework elegantly overcomes all the mentioned challenges. The significance of the reported research is multi-fold. First, the framework is valuable for real-time monitoring of thermal regulatory systems because of rapid forecasting. Second, researchers can address complex thermoregulation designs inasmuch as the approach is mesh-less. Finally, the framework facilitates systematic parameter identification and inverse modeling studies, perhaps the current framework's most significant utility.

S. Karimi, K. B. Nakshatrala

Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time-steps (multi-time-step coupling), and different time integrators and different numerical formulations (mixed methods) in different regions of the computational domain. We present two new monolithic multi-time-step mixed coupling methods for first-order transient systems. We shall employ unsteady advection-diffusion-reaction equation with linear decay as the model problem, which offers several unique challenges in terms of non-self-adjoint spatial operator and rich features in the solutions. We shall employ the dual Schur domain decomposition technique to handle the decomposition of domain into subdomains. Two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting. A systematic theoretical analysis (which includes numerical stability, influence of perturbations, bounds on drift along the subdomain interface) will be performed. The first coupling method ensures that there is no drift along the subdomain interface but does not facilitate explicit/implicit coupling. The second coupling method allows explicit/implicit coupling with controlled (but non-zero) drift in the solution along the subdomain interface. Several canonical problems will be solved to numerically verify the theoretical predictions, and to illustrate the overall performance of the proposed coupling methods. Finally, we shall illustrate the robustness of the proposed coupling methods using a multi-time-step transient simulation of a fast bimolecular advective-diffusive-reactive system.

M. S. Joshaghani, S. H. S. Joodat, K. B. Nakshatrala

Modeling flow through porous media with multiple pore-networks has now become an active area of research due to recent technological endeavors like geological carbon sequestration and recovery of hydrocarbons from tight rock formations. Herein, we consider the double porosity/permeability (DPP) model, which describes the flow of a single-phase incompressible fluid through a porous medium exhibiting two dominant pore-networks with a possibility of mass transfer across them. We present a stable mixed discontinuous Galerkin (DG) formulation for the DPP model. The formulation enjoys several attractive features. These include: (i) Equal-order interpolation for all the field variables (which is computationally the most convenient) is stable under the proposed formulation. (ii) The stabilization terms are residual-based, and the stabilization parameters do not contain any mesh-dependent parameters. (iii) The formulation is theoretically shown to be consistent, stable, and hence convergent. (iv) The formulation supports non-conforming discretizations and distorted meshes. (v) The DG formulation has improved element-wise (local) mass balance compared to the corresponding continuous formulation. (vi) The proposed formulation can capture physical instabilities in coupled flow and transport problems under the DPP model.

D. Z. Turner, K. B. Nakshatrala, M. J. Martinez

In this paper, we consider the flow of an incompressible fluid in a deformable porous solid. We present a mathematical model using the framework offered by the theory of interacting continua. In its most general form, this framework provides a mechanism for capturing multiphase flow, deformation, chemical reactions and thermal processes, as well as interactions between the various physics in a conveniently implemented fashion. To simplify the presentation of the framework, results are presented for a particular model than can be seen as an extension of Darcy's equation (which assumes that the porous solid is rigid) that takes into account elastic deformation of the porous solid. The model also considers the effect of deformation on porosity. We show that using this model one can recover identical results as in the framework proposed by Biot and Terzaghi. Some salient features of the framework are as follows: (a) It is a consistent mixture theory model, and adheres to the laws and principles of continuum thermodynamics, (b) the model is capable of simulating various important phenomena like consolidation and surface subsidence, and (c) the model is amenable to several extensions. We also present numerical coupling algorithms to obtain coupled flow-deformation response. Several representative numerical examples are presented to illustrate the capability of the mathematical model and the performance of the computational framework.

S. Srinivasan, K. B. Nakshatrala

In this paper, we present a stabilized mixed formulation for unsteady Brinkman equation. The formulation is systematically derived based on the variational multiscale formalism and the method of horizontal lines. The derivation does not need the assumption that the fine-scale variables do not depend on the time, which is the case with the conventional derivation of multiscale stabilized formulations for transient mixed problems. An expression for the stabilization parameter is obtained in terms of a bubble function, and appropriate bubble functions for various finite elements are also presented. Under the proposed formulation, equal-order interpolation for the velocity and pressure (which is computationally the most convenient) is stable. Representative numerical results are presented to illustrate the performance of the proposed formulation. Spatial and temporal convergence studies are also performed, and the proposed formulation performed well.

D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad

The following work presents a generalized (extended) finite element formulation for the advection-diffusion equation. Using enrichment functions that represent the exponential nature of the exact solution, smooth numerical solutions are obtained for problems with steep gradients and high Peclet numbers (up to Pe = 25) in one and two-dimensions. As opposed to traditional stabilized methods that require the construction of stability parameters and stabilization terms, the present work avoids numerical instabilities by improving the classical Galerkin solution with an enrichment function. To contextualize this method among other stabilized methods, we show by decomposition of the solution (in a multiscale manner) an equivalence to both Galerkin/least-squares type methods and those that use bubble functions. This work also presents a strategy for constructing the enrichment function for problems with complex geometries by employing a global-local approach.

D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad

The following paper compares a consistent Newton-Raphson and fixed-point iteration based solution strategy for a variational multiscale finite element formulation for incompressible Navier-Stokes. The main contributions of this work include a consistent linearization of the Navier-Stokes equations, which provides an avenue for advanced algorithms that require origins in a consistent method. We also present a comparison between formulations that differ only in their linearization, but maintain all other equivalences. Using the variational multiscale concept, we construct a stabilized formulation (that may be considered an extension of the MINI element to nonlinear Navier-Stokes). We then linearize the problem using fixed-point iteration and by deriving a consistent tangent matrix for the update equation to obtain the solution via Newton-Raphson iterations. We show that the consistent formulation converges in fewer iterations, as expected, for several test problems. We also show that the consistent formulation converges for problems for which fixed-point iteration diverges. We present the results of both methods for problems of Reynold's number up to 5000.

M. Shabouei, K. B. Nakshatrala

This paper presents a new approach to verify accuracy of computational simulations. We develop mathematical theorems which can serve as robust a posteriori error estimation techniques to identify numerical pollution, check the performance of adaptive meshes, and verify numerical solutions. We demonstrate performance of this methodology on problems from flow thorough porous media. However, one can extend it to other models. We construct mathematical properties such that the solutions to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties include the total minimum mechanical power, minimum dissipation theorem, reciprocal relation, and maximum principle for the vorticity. All the developed theorems have firm mechanical bases and are independent of numerical methods. So, these can be utilized for solution verification of finite element, finite volume, finite difference, lattice Boltzmann methods and so forth. In particular, we show that, for a given set of boundary conditions, Darcy velocity has the minimum total mechanical power of all the kinematically admissible vector fields. We also show that a similar result holds for Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and Darcy-Brinkman velocities have the minimum total dissipation among their respective kinematically admissible vector fields. Using numerical examples, we show that the minimum dissipation and total mechanical power theorems can be utilized to identify pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman equations are shown to satisfy a reciprocal relation, which has the potential to identify errors in the numerical implementation of boundary conditions.

K. B. Nakshatrala, D. Z. Turner

In this paper we consider a modification to Darcy equation by taking into account the dependence of viscosity on the pressure. We present a stabilized mixed formulation for the resulting governing equations. Equal-order interpolation for the velocity and pressure is considered, and shown to be stable (which is not the case under the classical mixed formulation). The proposed mixed formulation is tested using a wide variety of numerical examples. The proposed formulation is also implemented in a parallel setting, and the performance of the formulation for large-scale problems is illustrated using a representative problem. Two practical and technologically important problems, one each on enhanced oil recovery and geological carbon-dioxide sequestration, are solved using the proposed formulation. The numerical examples show that the predictions based on Darcy model are qualitatively and quantitatively different from that of the predictions based on the modified Darcy model, which takes into account the dependence of the viscosity on the pressure. In particular, the numerical example on the geological carbon-dioxide sequestration shows that Darcy model over-predicts the leakage into an abandoned well when compared to that of the modified Darcy model. On the other hand, the modified Darcy model predicts higher pressures and higher pressure gradients near the injection well. These predictions have dire consequences in predicting damage and fracture zones, and designing the seal, whose integrity is crucial to the safety of a geological carbon-dioxide sequestration geosystem.

V. S. Maduru, K. B. Nakshatrala

Accurate modeling of gas flow through porous media is critical for many technological applications, including reservoir performance prediction, carbon capture and sequestration, and fuel cells and batteries. However, such modeling remains challenging due to strong nonlinear behavior and uncertainty in model parameters. In particular, gas slippage effects described by the Klinkenberg model introduce pressure-dependent permeability, which complicates numerical simulation and obscures deviations from classical Darcy flow behavior. To address these challenges, we present an integrated modeling framework for gas transport in porous media that combines a Klinkenberg-enhanced constitutive relation, Hopf-Cole-transformed mixed-form linear governing equations, a shared-trunk neural network architecture, and a Deep Least-Squares (DeepLS) solver. The Hopf-Cole transformation reformulates the original nonlinear flow equations into an equivalent linear system closely related to the Darcy model, while the mixed formulation, together with a shared-trunk neural architecture, enables simultaneous and accurate prediction of both pressure and velocity fields. A rigorous convergence analysis is performed both theoretically and numerically, establishing the stability and convergence properties of the proposed solver. Importantly, the proposed framework also naturally facilitates inverse modeling of pressure-dependent permeability and slippage parameters from limited or indirect observations, enabling efficient estimation of flow properties that are difficult to measure experimentally. Numerical results demonstrate accurate recovery of flow dynamics and parameters across a wide range of pressure regimes, highlighting the framework's robustness, accuracy, and computational efficiency for gas transport modeling and inversion in tight formations.

D. Z. Turner, K. B. Nakshatrala, P. K. Notz

The following work compares two popular mixed finite elements used to model subsurface flow and transport in heterogeneous porous media; the lowest order Raviart-Thomas element and the variational multiscale stabilized element. Comparison is made based on performance for several problems of engineering relevance that involve highly heterogenous material properties (permeability ratios of up to $1\times10^5$), open flow boundary conditions (pressure driven flows), and large scale domains in two dimensions. Numerical experiments are performed to show the degree to which mass conservation is violated when a flow field computed using either element is used as the advection velocity in a transport model. The results reveal that the variational multiscale element shows considerable mass production or loss for problems that involve flow tangential to layers of differing permeability, but marginal violation of local mass balance for problems of less orthogonality in the permeability. The results are useful in establishing rudimentary estimates of the error produced by using the variational mutliscale element for several different types of problems.

V. S. Maduri, K. B. Nakshatrala

Porous materials -- natural or engineered -- often exhibit dual pore-network structures that govern processes such as mineral exploration and hydrocarbon recovery from tight shales. Double porosity/permeability (DPP) mathematical models describe incompressible fluid flow through two interacting pore networks with inter-network mass exchange. Despite significant advances in numerical methods, there remains a need for computational frameworks that enable rapid forecasting, data assimilation, and reliable inverse analysis. To address this, we present a physics-informed neural network (PINN) framework for forward and inverse modeling of DPP systems. The proposed approach encodes the governing equations in mixed form, along with boundary conditions, directly into the loss function, with adaptive weighting strategies to balance their contributions. Key features of the framework include adaptive weight tuning, dynamic collocation point selection, and the use of shared trunk neural architectures to efficiently capture the coupled behavior of the dual pore networks. It is inherently mesh-free, making it well-suited for complex geometries typical of porous media. It accurately captures discontinuities in solution fields across layered domains without introducing spurious oscillations commonly observed in classical finite element formulations. Importantly, the framework is well-suited for inverse analysis, enabling robust parameter identification in scenarios where key physical quantities -- such as the mass transfer coefficient in DPP models -- are difficult to measure directly. In addition, a systematic convergence analysis is provided to rigorously assess the stability, accuracy, and reliability of the method. The effectiveness and computational advantages of the approach are demonstrated through a series of representative numerical experiments.

N. V. Jagtap, M. K. Mudunuru, K. B. Nakshatrala

Prediction and control of chemical mixing are vital for many scientific areas such as subsurface reactive transport, climate modeling, combustion, epidemiology, and pharmacology. Due to the complex nature of mixing in heterogeneous and anisotropic media, the mathematical models related to this phenomenon are not analytically tractable. Numerical simulations often provide a viable route to predict chemical mixing accurately. However, contemporary modeling approaches for mixing cannot utilize available spatial-temporal data to improve the accuracy of the future prediction and can be compute-intensive, especially when the spatial domain is large and for long-term temporal predictions. To address this knowledge gap, we will present in this paper a deep-learning (DL) modeling framework applied to predict the progress of chemical mixing under fast bimolecular reactions. This framework uses convolutional neural networks (CNN) for capturing spatial patterns and long short-term memory (LSTM) networks for forecasting temporal variations in mixing. By careful design of the framework -- placement of non-negative constraint on the weights of the CNN and the selection of activation function, the framework ensures non-negativity of the chemical species at all spatial points and for all times. Our DL-based framework is fast, accurate, and requires minimal data for training.

S. Karimi, K. B. Nakshatrala

The lattice Boltzmann method (LBM) has established itself as a valid numerical method in computational fluid dynamics. Recently, multiple-relaxation-time LBM has been proposed to simulate anisotropic advection-diffusion processes. The governing differential equations of advective-diffusive systems are known to satisfy maximum principles, comparison principles, the non-negative constraint, and the decay property. In this paper, it will be shown that current single- and multiple-relaxation-time lattice Boltzmann methods fail to preserve these mathematical properties for transient diffusion-type equations. It will also be shown that the discretization of Dirichlet boundary conditions will affect the performance of lattice Boltzmann methods in meeting these mathematical principles. A new way of discretizing the Dirichlet boundary conditions is also proposed. Several benchmark problems have been solved to illustrate the performance of lattice Boltzmann methods and the effect of discretization of boundary conditions with respect to the aforementioned mathematical properties for transient diffusion and advection-diffusion equations.

K. B. Nakshatrala, A. J. Valocchi

In this research note we provide a variational basis for the optimal artificial diffusion method, which has been a cornerstone in developing many stabilized methods. The optimal artificial diffusion method produces exact nodal solutions when applied to one-dimensional problems with constant coefficients and forcing function. We first present a variational principle for a multi-dimensional advective-diffusive system, and then derive a new stable weak formulation. When applied to one-dimensional problems with constant coefficients and forcing function, this resulting weak formulation will be equivalent to the optimal artificial diffusion method. We present representative numerical results to corroborate our theoretical findings.

S. Darbha, K. B. Nakshatrala, K. R. Rajagopal

In this paper, we consider the vibratory motions of lumped parameter systems wherein the components of the system cannot be described by constitutive expressions for the force in terms of appropriate kinematical quantities. Such physical systems reduce to a system of differential-algebraic equations, which invariably need to be solved numerically. To illustrate the issues with clarity, we consider a simple system in which the dashpot is assumed to contain a "Bingham" fluid for which one cannot describe the force in the dashpot as a function of the velocity. On the other hand, one can express the velocity as a function of the force.

K. B. Nakshatrala, K. R. Rajagopal

In this paper we consider modifications to Darcy's equation wherein the drag coefficient is a function of pressure, which is a realistic model for technological applications like enhanced oil recovery and geological carbon sequestration. We first outline the approximations behind Darcy's equation and the modifications that we propose to Darcy's equation, and derive the governing equations through a systematic approach using mixture theory. We then propose a stabilized mixed finite element formulation for the modified Darcy's equation. To solve the resulting nonlinear equations we present a solution procedure based on the consistent Newton-Raphson method. We solve representative test problems to illustrate the performance of the proposed stabilized formulation. One of the objectives of this paper is also to show that the dependence of viscosity on the pressure can have a significant effect both on the qualitative and quantitative nature of the solution.