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A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods
Jerome Droniou, Robert Eymard, Thierry Gallouët, Raphaele Herbin
TL;DR
The paper addresses the lack of a common analysis for three discretizations of anisotropic heterogeneous diffusion on general grids. It generalizes their local constructions, proves algebraic equivalence, and relates the unified framework to finite element schemes. The framework can inherit mathematical properties of the original methods and yields an explicit lifting operator, while the method definition and convergence discussion retain stated assumptions for the linear diffusion setting.
Problem
The Hybrid, Mimetic and Mixed methods have similar behavior but lacked a common comparison because their analyses relied on different mathematical tools.
Method
The paper introduces generalized formulations and compares their local operators, inner products, fluxes and stabilization parameters to construct a unified framework.
Results
The Generalized Mimetic, Generalized Hybrid and Modified Mixed methods are identical for corresponding parameter choices.
Takeaways & Limitations
The unified framework extends mathematical properties of the initial methods and connects them with nonconforming and mixed finite element schemes through an explicit lifting operator.
Takeaways & Limitations
The paper defines and studies the methods for a linear single diffusion equation, while convergence requires additional conditions such as the stated discrete assumptions.
Abstract
from arXiv · showhide
We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.
1 Introduction
The paper studies three discretizations of anisotropic heterogeneous diffusion on general grids whose similar behavior motivates a unified analysis. It proves their equivalence and relates the resulting framework to other classical methods.
- Motivation: The Hybrid, Mimetic and Mixed methods discretize diffusion on general polygonal partitions using unknowns for the solution and edge fluxes.Their mathematical analyses use different tools, which had left their relationship insufficiently compared.
- Unified framework: The paper proves that the three resulting methods are identical up to suitable generalizations.The unified formulation can inherit mathematical properties established for the initial methods.
- Connections: The unified method is compared with nonconforming and mixed finite element schemes, yielding an explicit flux lifting operator.The lifting operator arises as a by-product for a particular choice of stabilization parameters.
- Scope: For more complex problems, the paper treats the linear diffusion equation as the source of the main ideas used to study those methods.Examples mentioned include incompressible Navier–Stokes, p-Laplacian, and nonlinear coupled problems.
2 The methods
The three methods use cell and edge solution unknowns together with conservative fluxes, while differing in how discrete gradients, inner products, and stabilization are constructed. Generalized formulations introduce flexible cell points and positive-definite stabilization parameters.
- Mixed method: The Mixed method uses the same cell and edge solution unknowns as the Hybrid method but defines the discrete gradient from fluxes instead of solution values.Its fluxes satisfy the shared conservativity structure on control-volume edges.
- Common structure: All three methods seek approximations of the solution and gradient fluxes on general polygonal grids.Finite volume conservativity and mimetic continuity impose a shared condition on interior-edge fluxes.
- Mimetic method: The Mimetic method defines a discrete divergence and its adjoint discrete flux operator through local flux inner products.The local matrices are symmetric positive definite and satisfy a discrete Stokes formula.
- Mimetic method: The Generalized Mimetic method freely chooses a point inside each cell and stabilization matrices, allowing the cell point to differ from the center of gravity.The generalized matrix construction preserves a corresponding generalized inner-product condition.
- Hybrid method: The Hybrid method adds edge solution unknowns, computes a consistent cellwise discrete gradient, and augments it with stabilization.The stabilization is introduced to obtain a coercive bilinear form, while the flux equation includes cell balance.
3 Algebraic correspondence between the three methods
The paper establishes an algebraic correspondence among generalized Mimetic, Hybrid and Modified Mixed formulations. Hybridization introduces common edge unknowns, enabling the methods to be compared through local inner products and flux relations.
- Main equivalence theorem: The main theorem states that the Generalized Mimetic, Generalized Hybrid and Modified Mixed methods are identical for corresponding parameter choices.The result is algebraic: each method’s parameters can be matched to parameters of the other two to produce the same scheme.
- Proof strategy: The proof proceeds by hybridizing the Generalized Mimetic method and then comparing it separately with Modified Mixed and Generalized Hybrid formulations.The three comparison stages are organized across Sections 3.1–3.3.
- Mimetic hybridization: Hybridization assigns edge values shared by neighboring cells and reproduces the boundary condition on boundary edges.The resulting formulation uses cell and edge unknowns together with conservative fluxes.
- Mimetic–Mixed correspondence: The Mimetic and Modified Mixed schemes coincide when their local flux inner products are chosen to be equal.The comparison reduces to matching the corresponding local matrix representations.
K TK(FK) and GT
The method correspondences are realized by matching low-dimensional local matrix representations and stabilization parameters. The resulting algebraic transformations show that Generalized Mimetic and Generalized Hybrid schemes represent the same family.
- Parameter correspondence: For any stabilization pair in the Generalized Mimetic method, an equivalent Generalized Hybrid stabilization matrix can be constructed, and conversely.The construction uses complementary subspaces associated with the local matrices.
- Matrix comparison: The local matrix identity is proved by separately matching the consistency and stabilization terms in the two formulations.The argument compares the terms of the Hybrid matrix representation with the inverse Mimetic matrix.
- Algebraic structure: The image of the reconstruction matrix equals the kernel of the local constraint mapping, providing the dimensional relation needed for the correspondence.The proof establishes inclusion and then equality by comparing ranks and dimensions.
- Conclusion: The Generalized Mimetic method is identical to the Generalized Hybrid method under corresponding parameter choices.The required algebraic computations occur in small-dimensional spaces and therefore have low practical cost.
K TK(·) and LK(·)T BH
The passage states that the quantities KLK(·) and LK(·) have trivial relations, referenced through equations (3.36) and (3.42).
- KLK(·) and LK(·) are the quantities discussed in this passage.
- Their relations are described as trivial.
- The stated relations are referenced through equations (3.36) and (3.42).
4 Convergence and error estimates
The HMMF framework unifies generalized Mimetic, Hybrid Finite Volume, and Modified Mixed Finite Volume schemes, then establishes convergence and error estimates under mesh and stability conditions.
- The generalized Mimetic, Hybrid FV, and Modified Mixed FV schemes are one HMMF family.
- The convergence analysis allows bounded measurable, uniformly elliptic coefficients and solutions only in H^1_0(Ω).
- Theorem 4.1 makes three framework-specific conditions equivalent and proves L2 convergence on suitable star-shaped polygonal meshes.
- Under usual mimetic mesh assumptions and Condition (S1), Generalized Mimetic solutions converge in L2 as mesh size tends to zero.
- The convergence proofs use discrete compactness, and the reconstructed gradient can converge strongly in L2.
- Order 1 flux and L2 solution error estimates transfer from Mimetic schemes to Generalized Hybrid and Modified Mixed methods under stated conditions.
- Order 2 L2 convergence is obtained when stabilization terms are sufficiently large, although a practical lower bound is not established.
5 Links with other methods
The HMMF can be interpreted, under suitable parameter choices, as nonconforming or mixed finite elements and as the classical two-point flux finite volume method. These links also yield convergence extensions and an explicit flux lifting operator, while two-point flux equivalence requires particular mesh and parameter conditions.
- Under one of three forms and parameter choices, the HMMF becomes a nonconforming finite element, mixed finite element, or two-point flux finite volume scheme.
- Nonconforming Finite Element method: A piecewise linear reconstruction on cones links the hybrid HMMF presentation to a nonconforming finite element formulation.
- Nonconforming Finite Element method: Convergence properties extend to the nonconforming formulation because the differing right-hand sides are separated by a term of order h.
- Mixed Finite Element methods: The mixed HMMF formulation provides a mixed finite element method on general meshes, distinct from the Raviart–Thomas RT0 method even on simplicial meshes.
- Mixed Finite Element methods: The associated lifting reconstructs an interior-cell flux from boundary fluxes, preserving boundary normal fluxes, a divergence relation, and constant vector fields.
- Two-point flux cases: For isotropic diffusion, the standard two-point flux formula with harmonic averaging is recovered in the relevant super-admissible discretizations and parameter choices.
6 Appendix
The appendix establishes technical foundations for the generalized Mimetic definition, including admissible weight functions, equivalent matrix and inner-product conditions, and an abstract inner-product transfer result.
- The matrix equivalence is established in both directions, including symmetry and positive definiteness of the resulting reduced matrix.
- About the Generalized Mimetic definition: An affine weight function satisfying the required conditions exists for every bounded non-empty open cell and any selected point x_E.
- About the Generalized Mimetic definition: The generalized local inner-product condition is equivalent to the matrix representation M_E satisfying the corresponding factorization condition.
- Linear maps with identical kernels can transfer an inner product from one image space to the other while preserving pairwise products of mapped vectors.
- The proof gives an explicit construction by choosing complements, inverting one map on a complement, and extending the induced inner product.