Name
Abstract | ||
|---|---|---|
The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems. |
Year | DOI | Venue |
|---|---|---|
2014 | 10.1016/j.jcp.2013.07.031 | J. Comput. Physics |
Keywords | Field | DocType |
major mimetic idea,polyhedral mesh,mimetic approximation,diffusion problem,exact mathematical identity,mimetic finite difference method,discrete maximum principle,mimetic finite difference,various discrete operator,major convergence result,mimetic methodology | Mathematical optimization,Finite difference,Physical system,Electromagnetics,Duality (optimization),Operator (computer programming),Tensor calculus,Conservation law,Mathematics,Numerical stability | Journal |
Volume | ISSN | Citations |
257 | 0021-9991 | 15 |
PageRank | References | Authors |
1.06 | 48 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| K. Lipnikov | 1 | 521 | 57.35 |
| Gianmarco Manzini | 2 | 239 | 27.46 |
| Mikhail Shashkov | 3 | 542 | 54.19 |