The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials
Journal of Computational Physics
A local support-operators diffusion discretization scheme for quadrilateral r-z meshes
Journal of Computational Physics
Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle
Mathematics of Computation
A tensor artificial viscosity using a mimetic finite difference algorithm
Journal of Computational Physics
Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes
Journal of Computational Physics
Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Monotonicity of control volume methods
Numerische Mathematik
A weak discrete maximum principle for hp-FEM
Journal of Computational and Applied Mathematics
On discrete maximum principles for nonlinear elliptic problems
Mathematics and Computers in Simulation
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
Journal of Computational Physics
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
A monotone finite volume method for advection-diffusion equations on unstructured polygonal meshes
Journal of Computational Physics
Discrete maximum principle for parabolic problems solved by prismatic finite elements
Mathematics and Computers in Simulation
The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes
Journal of Computational Physics
A Mimetic Discretization of the Stokes Problem with Selected Edge Bubbles
SIAM Journal on Scientific Computing
Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes
SIAM Journal on Numerical Analysis
An improved monotone finite volume scheme for diffusion equation on polygonal meshes
Journal of Computational Physics
Journal of Computational Physics
Mimetic finite difference method
Journal of Computational Physics
Mimetic scalar products of discrete differential forms
Journal of Computational Physics
Hi-index | 31.47 |
The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart-Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.