Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes

Lipnikov, Konstantin; Manzini, Gianmarco

doi:10.1007/978-3-319-41640-3_10

Cited by 3 publications

(3 citation statements)

References 57 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Examples are found in . A more exhaustive list of contributions to the development of the MFD method can also be found in the two review works [33] and [34]. Other discretization frameworks related to general meshes include the polygonal/polyhedral finite element method (PFEM) [35,36], the finite volume methods [37], the hybrid high-order (HHO) method [38,39], the discontinuous Galerkin (DG) method [40,41], the hybridized discontinuous Galerkin (HDG) method [42], the weak Galerkin (wG) method [43], the virtual element method (VEM) [44] and its connections with the PFEM [45], and the more recent Discontinuous Skeletal Gradient Discretization (DSGD) [46].…”

Section: Introductionmentioning

confidence: 99%

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

2019

Self Cite

View full text Add to dashboard Cite

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approximation of the divergence and gradient operators and the two inner products for the discrete analogs of fluxes and scalar unknowns. The discrete divergence and gradient operators are built according to a discrete duality relation. The inner product for the flux grid functions is built by explicitly imposing the conditions of consistency and stability. The family of semi-discrete mimetic methods is proved theoretically to be energy-stable as the corresponding continuous problem. Then, a full discretization is derived by combining via MOL the MFD method of order k with time marching schemes from the backward differentiation formula of order k + 2. Optimal order of accuracy is demonstrated for the scalar variable and verified numerically by solving the time-dependent diffusion problems with a variable diffusion tensor for k from 0 to 3 on three different unstructured mesh families.

show abstract

Section: Introductionmentioning

confidence: 99%

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

2019

Self Cite

View full text Add to dashboard Cite

show abstract

“…The scheme used in this work chooses the divergence as the primary operator and uses finitevolume-like approximation of it in cells, while the approximation of the gradient operator is built in such a way that it is negatively adjoint to the discrete divergence in appropriate discrete spaces with their scalar products. Details on the discrete gradient construction can be found in [34]. The presence of solution-dependent relative permeability in (5) makes the construction of the scheme more complicated, since some approximations of nonlinear coefficient may lead to incorrect solutions for transient equations [35].…”

Section: Mimetic Finite Difference Schemementioning

confidence: 99%

First-order continuation method for steady-state variably saturated groundwater flow modeling

Anuprienko¹

2022

Preprint

View full text Add to dashboard Cite

Recently, the nonlinearity continuation method has been used to numerically solve boundary value problems for steady-state Richards equation. The method can be considered as a predictor-corrector procedure with the simplest form which has been applied to date having a trivial, zeroth-order predictor. In this article, effect of a more sophisticated predictor technique is examined. Numerical experiments are performed with finite volume and mimetic finite difference discretizations on various problems, including reallife examples.

show abstract

“…• general presentation of the MFD method:book [31]; review papers [97,108]; connection of MFD with other methods [105]; benchmarks [103,111]; conference paper [112],…”

Section: 5mentioning

confidence: 99%

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

Manzini

2017

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Discretization of Mixed Formulations of Elliptic Problems on Polyhedral Meshes

Cited by 3 publications

References 57 publications

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

First-order continuation method for steady-state variably saturated groundwater flow modeling

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

Contact Info

Product

Resources

About