Which are Better Conditioned Meshes Adaptive, Uniform, Locally Refined or Localised

Adaptive, locally refined and locally adjusted meshes are preferred over uniform meshes for capturing singular or localised solutions. Roughly speaking, for a given degree of freedom a solution associated with adaptive, locally refined and locally adjusted meshes is more accurate than the solution given by uniform meshes. In this work, we answer the question which meshes are better conditioned. We found, for approximately same degree of freedom (same size of matrix), it is easier to solve a system of equations associated with an adaptive mesh.Comment: 4 Page

A Comparison of Consistent Discretizations for Elliptic Problems on Polyhedral Grids

In this work, we review a set of consistent discretizations for second-order elliptic equations, and compare and contrast them with respect to accuracy, monotonicity, and factors affecting their computational cost (degrees of freedom, sparsity, and condition numbers). Our comparisons include the linear and nonlinear TPFA method, multipoint flux-approximation (MPFA-O), mimetic methods, and virtual element methods. We focus on incompressible flow and study the effects of deformed cell geometries and anisotropic permeability.acceptedVersio

From face to element unknowns by local static condensation with application to nonconforming finite elements

International audienceWe derive in this paper a new local static condensation strategy which allows to reduce significantly the number of unknowns in algebraic systems arising in discretization of partial differential equations. We apply it to the discretization of a model linear elliptic diffusion and a model nonlinear parabolic advection--diffusion--reaction problem by Crouzeix--Raviart nonconforming finite elements. Herein, the unknowns, originally associated with the mesh faces, can be reduced to new unknowns associated with the mesh elements. The resulting matrices are sparse, with possibly only four nonzero entries per row in two space dimensions, positive definite in dependence on the mesh geometry and the diffusion--dispersion tensor, but in general nonsymmetric. Our approach consists in introducing new element unknowns, the identification of suitable local vertex-based subproblems, and the inversion of the corresponding local matrices. We give sufficient conditions for the well-posedness of the local problems, as well as for the resulting global one. In addition, we provide a geometrical interpretation which suggests how to influence the form of the local and global matrices depending on the local mesh and data. We finally present an abstract generalization allowing for a further reduction of the number of unknowns, typically to one unknown per a set of mesh elements. We conclude by numerical experiments which show that the condition number of the resulting matrices is robust with respect to the mesh anisotropies and the diffusion tensor inhomogeneities

Modeling and discretization of flow in porous media with thin, full-tensor permeability inclusions

When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower-dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full-tensor permeability effects also in the out-of-plane direction of the inclusion. The governing equations of flow for the lower-dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed-dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two-dimensional and three-dimensional.publishedVersio