Mathematical modeling of a microwave plasma generator
USSR Computational Mathematics and Mathematical Physics
Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux
Journal of Computational Physics
Numerical modelling of two-dimensional gas-dynamic flows on a variable-structure mesh
USSR Computational Mathematics and Mathematical Physics
USSR Computational Mathematics and Mathematical Physics
USSR Computational Mathematics and Mathematical Physics
Difference schemes for the Navier-Stokes equations on a net consisting of Dirichlet cells
USSR Computational Mathematics and Mathematical Physics
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Axisymmetric difference operators in an orthogonal coordinate system
USSR Computational Mathematics and Mathematical Physics
Direct discretization of planar div-curl problems
SIAM Journal on Numerical Analysis
Computational methods in Lagrangian and Eulerian hydrocodes
Computer Methods in Applied Mechanics and Engineering
Support-operator finite-difference algorithms for general elliptic problems
Journal of Computational Physics
Summation by parts, projections, and stability. I
Mathematics of Computation
Summation by parts, projections, and stability. II
Mathematics of Computation
Chain models and finite elements analysis: an executable CHAINS formulation of plane stress
Computer Aided Geometric Design - Special issue on grid generation, finite elements, and geometric design
Solving diffusion equations with rough coefficients in rough grids
Journal of Computational Physics
The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
Journal of Computational Physics
Journal of Computational Physics
Applied Numerical Mathematics
Covolume Solutions of Three-Dimensional Div-Curl Equations
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Convergence analysis of a covolume scheme for Maxwell's equations in three dimensions
Mathematics of Computation
A local support-operators diffusion discretization scheme for quadrilateral r-z meshes
Journal of Computational Physics
Formulations of artificial viscosity for multi-dimensional shock wave computations
Journal of Computational Physics
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
The construction of compatible hydrodynamics algorithms utilizing conservation of total energy
Journal of Computational Physics
Journal of Computational Physics
Mimetic discretizations for Maxwell's equations
Journal of Computational Physics
Canonical construction of finite elements
Mathematics of Computation
Conservation properties of unstructured staggered mesh schemes
Journal of Computational Physics
A finite volume method for the approximation of diffusion operators on distorted meshes
Journal of Computational Physics
A multivector data structure for differential forms and equations
Mathematics and Computers in Simulation
A tensor artificial viscosity using a mimetic finite difference algorithm
Journal of Computational Physics
The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods
SIAM Journal on Numerical Analysis
Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation
SIAM Journal on Numerical Analysis
Summation by parts operators for finite difference approximations of second derivatives
Journal of Computational Physics
Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes
Journal of Computational Physics
Mathematics and Computers in Simulation - Special issue: Applications of computer algebra in science, engineering, simulation and special software
SIAM Journal on Numerical Analysis
Steady-State Computations Using Summation-by-Parts Operators
Journal of Scientific Computing
Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals
SIAM Journal on Numerical Analysis
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes
SIAM Journal on Numerical Analysis
A mortar mimetic finite difference method on non-matching grids
Numerische Mathematik
SIAM Journal on Numerical Analysis
The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes
Journal of Computational Physics
A mixed finite volume scheme for anisotropic diffusion problems on any grid
Numerische Mathematik
Mesh locking effects in the finite volume solution of 2-D anisotropic diffusion equations
Journal of Computational Physics
Monotonicity of control volume methods
Numerische Mathematik
A Multipoint Flux Mixed Finite Element Method
SIAM Journal on Numerical Analysis
Higher-order mimetic methods for unstructured meshes
Journal of Computational Physics
Discrete calculus methods for diffusion
Journal of Computational Physics
SIAM Journal on Numerical Analysis
A residual based error estimator for the Mimetic Finite Difference method
Numerische Mathematik
A multilevel multiscale mimetic (M3) method for two-phase flows in porous media
Journal of Computational Physics
High-order mimetic finite difference method for diffusion problems on polygonal meshes
Journal of Computational Physics
A Higher-Order Formulation of the Mimetic Finite Difference Method
SIAM Journal on Scientific Computing
Local flux mimetic finite difference methods
Numerische Mathematik
Mimetic finite difference method for the Stokes problem on polygonal meshes
Journal of Computational Physics
Convergence analysis of the high-order mimetic finite difference method
Numerische Mathematik
Convergence Analysis of the Mimetic Finite Difference Method for Elliptic Problems
SIAM Journal on Numerical Analysis
The convergence of mimetic discretization for rough grids
Computers & Mathematics with Applications
Innovative mimetic discretizations for electromagnetic problems
Journal of Computational and Applied Mathematics
Isogeometric Analysis: Toward Integration of CAD and FEA
Isogeometric Analysis: Toward Integration of CAD and FEA
Journal of Computational Physics
The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes
Journal of Computational Physics
The Discrete Duality Finite Volume Method for Convection-diffusion Problems
SIAM Journal on Numerical Analysis
A Mimetic Discretization of the Stokes Problem with Selected Edge Bubbles
SIAM Journal on Scientific Computing
Journal of Computational Physics
Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes
SIAM Journal on Numerical Analysis
A mimetic discretization of the Reissner–Mindlin plate bending problem
Numerische Mathematik
A 3D Discrete Duality Finite Volume Method for Nonlinear Elliptic Equations
SIAM Journal on Scientific Computing
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
Mimetic scalar products of discrete differential forms
Journal of Computational Physics
Journal of Computational Physics
Hi-index | 31.45 |
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.