Mimetic finite difference approximation of quasilinear elliptic problems

Antonietti, Paola F.; Bigoni, Nadia; Verani, Marco

doi:10.1007/s10092-014-0107-y

Cited by 15 publications

(15 citation statements)

References 45 publications

(87 reference statements)

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“…We cite here, in particular, the two-dimensional Discrete Duality Finite Volume schemes studied in [4] (cf. also the precursor papers [1][2][3]), the Mixed Finite Volume scheme of [36] (inspired by [37]) valid in arbitrary space dimension, and the Mimetic Finite Difference method of [5] for p P p1, 2q and under more restrictive assumptions than (2.2). High-order discontinuous Galerkin approximations have also been considered in [22].…”

Section: Introductionmentioning

confidence: 99%

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

Pietro¹,

Droniou²

2016

Math. Comp.

131

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In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady nonlinear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, L p -stability and W s,p -approximation properties for L 2 -projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

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Section: Introductionmentioning

confidence: 99%

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

Pietro¹,

Droniou²

2016

Math. Comp.

131

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“…• 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

Manzini

2017

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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.

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“…More recently, non conforming numerical schemes defined on polytopal meshes were introduced; discrete duality finite volume schemes were studied in [3,4,2,1]. Other schemes which have been showed to be part of the gradient discretisation method reviewed in the recent book [18], were also studied for the Leray-Lions type problems, namely the SUSHI scheme [19], the mixed finite volume scheme [17], the mimetic finite difference method [5]; the discontinuous Galerkin approximation was considered in [14,20] and the hybrid high order scheme in [16]. In all these works, usually only one type of boundary conditions is considered (most often homogeneous Dirichlet boundary conditions).…”

Section: Introductionmentioning

confidence: 99%

A unified analysis of elliptic problems with various boundary conditions and their approximation

Droniou¹,

Eymard²,

Gallouët³

et al. 2019

Czech.Math.J.

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We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods.The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincaré inequalities and the surjectivity of the divergence operator in appropriate spaces.2 An illustrative example

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Mimetic finite difference approximation of quasilinear elliptic problems

Cited by 15 publications

References 45 publications

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

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

A unified analysis of elliptic problems with various boundary conditions and their approximation

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