Mimetic finite differences for nonlinear and control problems

2017

stiffness matrix is rank deficient). In the VEM we fix this issue by adding a stabilization term to the bilinear form. Stabilization terms are required not to upset the exactness property for polynomials and to scale properly with respect to the mesh size and the problem coefficients.1.4.1. The low-order setting for polygonal cells (in 2D) and polyhedral cells (in 3D) can be obtained as a straightforward generalization of the FEM. The high-order setting deserves special care in the treatment of variable coefficients in order not to loose the optimality of the discretization and in the 3D formulation.1.5. The shape functions are virtual in the sense that we never compute them explicitly (not even approximately) inside the elements. Note that the polynomial projection of the approximate solution is readily available from the degrees of freedom and we can use it as the numerical solution.1.6. The major advantage offered by the VEM is in the great flexibility in admitting unstructured polygonal and polyhedral meshes.1.7. The VEM can be seen as an evolution of the MFD method. Background material (and a few historical notes) will be given in the final section of this document.

“…• quasilinear elliptic problems [11], nonlinear and control problems [6,7], obstacle elliptic problem [10],…”

Section: 5mentioning

confidence: 99%

Annotations on the virtual element method for second-order elliptic problems

2017

“…h,c and a (2) h,c are the consistency and stability terms (to be defined later in the section), and operator div h is defined cell-by-cell as the local L 2 -orthogonal projection of the continuum divergence operator onto C h . The virtual space SF c includes polynomial as well as nonpolynomial functions.…”

Section: B)mentioning

confidence: 99%

“…Many ideas underpinning the MFD method were originally formulated in the sixties for orthogonal meshes using the finite difference framework from which the name of the method was derived. Over the years, the MFD method has been extensively developed for the solution of a wide range of scientific and engineering problems in continuum mechanics [39], electromagnetics [32,34], fluid flows [36,7,8,17,6], elasticity [5,10], obstacle and control problems [3,1,2], diffusion [33], discretization of differential forms [11,41,12], and eigenvalue analysis [15]. An extensive list of people who contributed to the development of the MFD method can be found in the recent book [9] and review paper [35].…”

Section: Introductionmentioning

confidence: 99%

Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes

Lipnikov

Lecture Notes in Computational Science and Engineering

2016

We review basic design principles underpinning the construction of mimetic finite difference and a few finite volume and finite element schemes for mixed formulations of elliptic problems. For a class of low-order mixed-hybrid schemes, we show connections between these principles and prove that the consistency and stability conditions must lead to a member of the mimetic family of schemes regardless of the selected discretization framework. Finally, we give two examples of using flexibility of the mimetic framework: derivation of higher-order schemes and convergent schemes for nonlinear problems with small diffusion coefficients.

“…We just recall here the books, 5,18,21 and the review papers. 1,2,7,13,14,16,19 In recent times the matter received an increasing attention, due to the combination of several factors that include the convenience in mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening, and the like. Recent developments include the evolution of Mimetic Finite Differences in the direction of nodal unknowns 9 or edge unknowns, 8 the connections with Finite Volumes methods (see e.g.…”

Section: Prefacementioning

confidence: 99%

Recent techniques for PDE discretizations on polyhedral meshes

Bellomo

Brezzi

Math. Models Methods Appl. Sci.

2014

This brief paper is an introduction to the papers published in a special issue devoted to survey on recent techniques for discretizing Partial Differential Equations on general polygonal and polyhedral meshes. The number of different techniques to deal with discretizations on polygonal and polyhedral meshes is quite huge, and their history is quite long. Here we concentrate on the most recent techniques, including Mimetic Finite Differences, Virtual Element Methods, and the recent developments, in this direction, of Finite Volumes and Discontinuous Galerkin Methods.