3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids

Abstract

We present a number of test cases and meshes that were designed as a benchmark for numerical schemes dedicated to the approximation of threedimensional anisotropic and heterogeneous diffusion problems. These numerical schemes may be applied to general, possibly non conforming, meshes composed of tetrahedra, hexahedra and quite distorted general polyhedra. A number of methods were tested among which conforming finite element methods, discontinuous Galerkin finite element methods, cell-centered finite volume methods, discrete duality finite volume methods, mimetic finite difference methods, mixed finite element methods, and gradient schemes. We summarize the results presented by the participants to the benchmark, which range from the number of unknowns, the approximation errors of the solution and its gradient, to the minimum and maximum values and energy. We also compare the performance of several iterative or direct linear solvers for the resolution of the linear systems issued from the presented schemes.

Small-stencil 3D schemes for diffusive flows in porous media

“…Several tests are taken from the FVCA Benchmark 3D [23]. Therein additional results on the VAG scheme are presented for a larger family of numerical tests and for several criteria of comparison.…”

“…An intense research activity has recently been devoted to the numerical approximation Φ K,σ of Φ K,σ (ū), particularly in the oil engineering framework, where such problems are of utmost importance; we refer for instance to the various schemes presented at the FVCA5 and FVCA6 benchmarks [23,25].…”

A domain-specific embedded language in C++ for lowest-order discretizations of diffusive problems on general meshes

“…When it comes to numerical performance, recent benchmarks [25,30] have pointed out that the choice of the scheme for a given problem should be driven by multiple factors including, e.g., the features of the problem itself (heterogeneity, presence of convection), the computational mesh (which may result from an upstream modeling process), and the required precision. In this respect, there is an increasing urge to dispose of libraries covering a wide range of lowest-order methods and applications based on similar experiences in the context of Finite Element (FE) methods.…”

“…A different approach based on the analogy between lowest order methods in variational formulation and discontinuous Galerkin methods has been proposed by one of the authors in [13][14][15] (Cell Centered Galerkin methods). The key advantage of this approach is that it largely benefits from the well-established theory for discontinuous Galerkin methods [17].When it comes to numerical performance, recent benchmarks [25,30] have pointed out that the choice of the scheme for a given problem should be driven by multiple factors including, e.g., …”

“…When it comes to numerical performance, recent benchmarks [25,30] have pointed out that the choice of the scheme for a given problem should be driven by multiple factors including, e.g.,…”