Miroslav Kuchta

Miroslav Kuchta, Magne Nordaas, Joris C. G. Verschaeve et al.

We study preconditioners for a model problem describing the coupling of two elliptic subproblems posed over domains with different topological dimension by a parameter dependent constraint. A pair of parameter robust and efficient preconditioners is proposed and analyzed. Robustness and efficiency of the preconditioners is demonstrated by numerical experiments.

Miroslav Kuchta, Kent-Andre Mardal, Mikael Mortensen

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work we consider a simplified model problem of a 3d-1d coupling and the main objective is to construct algorithms that may utilize stan- dard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well-defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the sys- tem. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.

Trygve Bærland, Miroslav Kuchta, Kent-Andre Mardal

Coupled multiphysics problems often give rise to interface conditions naturally formulated in fractional Sobolev spaces. Here, both positive- and negative fractionality are common. When designing efficient solvers for discretizations of such problems it would then be useful to have a preconditioner for the fractional Laplacian. In this work, we develop an additive multigrid preconditioner for the fractional Laplacian with positive fractionality, and show a uniform bound on the condition number. For the case of negative fractionality, we re-use the preconditioner developed for the positive fractionality and left-right multiply a regular Laplacian with a preconditioner with positive fractionality to obtain the desired negative fractionality. Implementational issues are outlined in details as the differences between the discrete operators and their corresponding matrices must be addressed when realizing these algorithms in code. We finish with some numerical experiments verifying the theoretical findings.

Miroslav Kuchta, Kent-Andre Mardal, Mikael Mortensen

The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a non-singular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well-posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, while a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. Based on this observation a modification to the conjugate gradient method is proposed that achieves optimal error convergence of the computed solution.

Karl Erik Holter, Miroslav Kuchta, Kent-André Mardal

We study perfusion by a multiscale model coupling diffusion in the tissue and diffusion along the one-dimensional segments representing the vasculature. We propose a block-diagonal preconditioner for the model equations and demonstrate its robustness by numerical experiments. We compare our model to a macroscale model by [P. Tofts, Modelling in DCE MRI, 2012].

Samuele Ferri, Chiara Giraudo, Valerio Loi et al.

In the present paper, we analyze in detail the spectral features of the matrix sequences arising from the Taylor-Hood $\mathbb{P}_2$-$\mathbb{P}_1$ approximation of variable viscosity for $2d$ Stokes problem under weak assumptions on the regularity of the diffusion. Localization and distributional spectral results are provided, accompanied by numerical tests and visualizations. A preliminary study of the impact of our findings on the preconditioning problem is also presented. A final section with concluding remarks and open problems ends the current work.

Esteban Henríquez, Miroslav Kuchta, Jeonghun J. Lee et al.

We propose parameter-robust preconditioners for the statically condensed linear system arising from a hybridizable discontinuous Galerkin discretization of the coupled Stokes--Darcy system. The design strategy relies on first applying the operator-preconditioning framework [Numer. Linear Algebra Appl., 18(1):1--40, 2011] to construct a preconditioner for the non-condensed discretization. This is done by proving uniform well-posedness of the scheme. Next, we prove robustness of the resulting condensed preconditioner applied to the reduced linear system using the framework we proposed in [SIAM J. Sci. Comput., 47(6):A3212--A3238, 2025]. Numerical examples demonstrate robustness of the proposed preconditioners.

Sebastian K. Mitusch, Simon W. Funke, Miroslav Kuchta

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks.