Garth N. Wells

Martin S. Alnaes, Anders Logg, Kristian B. Oelgaard et al.

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multi-field problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.

Anders Logg, Garth N. Wells

We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code.

Sander Rhebergen, Garth N. Wells

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.

Eveline Rosseel, Garth N. Wells

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. One-shot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby obviating the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented.

Sander Rhebergen, Garth N. Wells

Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise solenoidal. It is shown that the method is inf-sup stable for both equal-order and locally Taylor--Hood type spaces, and \emph{a priori} error estimates are developed. The considered method can be constructed to have the same global algebraic structure as a conforming Galerkin method, unlike standard discontinuous Galerkin methods that have greater number of degrees of freedom than conforming Galerkin methods on a given mesh. We assert that this method is amongst the simplest and most flexible finite element approaches for Stokes flow that provide local mass conservation. With this contribution the mathematical basis is established, and this supports the performance of the method that has been observed experimentally in other works.

Garth N. Wells

Analysis of an interface stabilised finite element method for the scalar advection-diffusion-reaction equation is presented. The method inherits attractive properties of both continuous and discontinuous Galerkin methods, namely the same number of global degrees of freedom as a continuous Galerkin method on a given mesh and the stability properties of discontinuous Galerkin methods for advection dominated problems. Simulations using the approach in other works demonstrated good stability properties with minimal numerical dissipation, and standard convergence rates for the lowest order elements were observed. In this work, stability of the formulation, in the form of an inf-sup condition for the hyperbolic limit and coercivity for the elliptic case, is proved, as is order $k+1/2$ order convergence for the advection-dominated case and order $k+1$ convergence for the diffusive limit in the $L^{2}$ norm. The analysis results are supported by a number of numerical experiments.

Robert Jan Labeur, Garth N. Wells

A hybrid method for the incompressible Navier--Stokes equations is presented. The method inherits the attractive stabilizing mechanism of upwinded discontinuous Galerkin methods when momentum advection becomes significant, equal-order interpolations can be used for the velocity and pressure fields, and mass can be conserved locally. Using continuous Lagrange multiplier spaces to enforce flux continuity across cell facets, the number of global degrees of freedom is the same as for a continuous Galerkin method on the same mesh. Different from our earlier investigations on the approach for the Navier--Stokes equations, the pressure field in this work is discontinuous across cell boundaries. It is shown that this leads to very good local mass conservation and, for an appropriate choice of finite element spaces, momentum conservation. Also, a new form of the momentum transport terms for the method is constructed such that global energy stability is guaranteed, even in the absence of a point-wise solenoidal velocity field. Mass conservation, momentum conservation and global energy stability are proved for the time-continuous case, and for a fully discrete scheme. The presented analysis results are supported by a range of numerical simulations.

Sander Rhebergen, Garth N. Wells

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners.

Steven Vandekerckhove, Garth N. Wells, Herbert De Gersem et al.

Matched layers are commonly used in numerical simulations of wave propagation to model (semi-)infinite domains. Attenuation functions describe the damping in layers, and provide a matching of the wave impedance at the interface between the domain of interest and the absorbing region. Selecting parameters in the attenuation functions is non-trivial. In this work, an optimisation procedure for automatically calibrating matched layers is presented. The procedure is based on solving optimisation problems constrained by partial differential equations with polynomial and piecewise-constant attenuation functions. We show experimentally that, for finite element time domain simulations, piecewise-constant attenuation function are at least as efficient as quadratic attenuation functions. This observation leads us to introduce consecutive matched layers as an alternative to perfectly matched layers, which can easily be employed for problems with arbitrary geometries. Moreover, the use of consecutive matched layers leads to a reduction in computational cost compared to perfectly matched layers. Examples are presented for acoustic, elastodynamic and electromagnetic problems. Numerical simulations are performed with the libraries FEniCS/DOLFIN and dolfin-adjoint, and the computer code to reproduce all numerical examples is made freely available.

Jack S. Hale, Lizao Li, Chris N. Richardson et al.

Containers are an emerging technology that hold promise for improving productivity and code portability in scientific computing. We examine Linux container technology for the distribution of a non-trivial scientific computing software stack and its execution on a spectrum of platforms from laptop computers through to high performance computing (HPC) systems. We show on a workstation and a leadership-class HPC system that when deployed appropriately there are no performance penalties running scientific programs inside containers. For Python code run on large parallel computers, the run time is reduced inside a container due to faster library imports. The software distribution approach and data that we present will help developers and users decide on whether container technology is appropriate for them. We also provide guidance for the vendors of HPC systems that rely on proprietary libraries for performance on what they can do to make containers work seamlessly and without performance penalty.