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Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces
Robert Eymard, Thierry Gallouët, Raphaele Herbin
TL;DR
The paper addresses heterogeneous anisotropic diffusion on general, possibly nonconforming meshes, seeking a scheme that remains close to cell-centred methods while conserving fluxes at highly heterogeneous interfaces. It develops SUSHI with selectively retained interface unknowns and establishes convergence, numerical accuracy, and error estimates under suitable regularity assumptions.
Problem
The paper addresses discretisation of heterogeneous anisotropic diffusion on general meshes while seeking cell-centred efficiency and local conservativity across highly heterogeneous interfaces.
Method
SUSHI uses control-volume-centre values together with selectively retained internal-interface unknowns, ranging from a cell-centred scheme to the hybrid scheme HFV.
Results
The approximate solution and discrete gradient converge in L2 to the continuous solution as the mesh size tends to zero, and numerical experiments report order 2 L2 convergence for the solution.
Takeaways & Limitations
The scheme supports heterogeneous anisotropic diffusion on distorted nonconforming meshes, with convergence established without regularity beyond u ∈ H1_0(Ω).
Takeaways & Limitations
Retaining interface unknowns can make the scheme expensive, although algebraic elimination is sometimes possible and lacks general invertibility or symmetry guarantees.
Abstract
from arXiv · showhide
A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.
1 Introduction
The paper targets heterogeneous anisotropic diffusion on distorted, possibly nonconforming meshes, where standard finite volume schemes may be difficult to adapt or analyze. It develops SUSHI to balance cell-centred efficiency, interface conservativity, symmetry, and convergence.
- Heterogeneous anisotropic diffusion arises in applications including hydrogeology, oil reservoir simulation, plasma physics, semiconductors, and biology.
- Standard finite difference, finite element, and finite volume schemes are not easily adapted to heterogeneous anisotropic diffusion operators.
- Engineering and computing constraints require methods that handle distorted and possibly nonconforming meshes without relying on restrictive geometric conditions.
- The scheme is designed to remain close to cell-centred discretisation while preserving symmetry and local flux conservativity at interfaces between highly heterogeneous media.
- SUSHI retains interface unknowns only when needed, reduces to a cell-centred scheme when none are kept, and becomes the hybrid scheme HFV when all internal edges are included.
- The paper evaluates practical properties numerically and develops convergence and error analyses for the resulting schemes.
2 Fundamentals for a class of nonconforming schemes
The paper designs SUSHI for heterogeneous anisotropic diffusion on general conforming or nonconforming meshes, balancing cell-centred efficiency, symmetry, conservativity, and convergence. It selectively retains interface unknowns near tensor discontinuities while eliminating others, yielding a less expensive scheme with convergence guarantees and error estimates under regularity.
- Desired properties: SUSHI targets polyhedral conforming or nonconforming grids in two or more dimensions, without requiring convex control volumes.The mesh may include generalized hexahedra whose nonplanar faces consist of several planar sub-faces.
- Desired properties: The design seeks sparse, symmetric positive systems that remain close to cell-centred schemes while preserving local flux conservativity across highly heterogeneous interfaces.These requirements combine computational considerations with accuracy at material interfaces.
- Hybrid and cell-centred formulations: Hybrid schemes use unknowns in control volumes and interfaces, but retaining all interface values increases the system size and computational expense.The unknown count is card(M) + card(E), although elimination may sometimes be possible on specialized meshes.
- Hybrid and cell-centred formulations: A cell-centred nonconforming approach is efficient for Λ = Id but can poorly approximate local fluxes at strongly heterogeneous interfaces, especially on coarse meshes.This motivates retaining selected interface degrees of freedom rather than eliminating them everywhere.
- Composite scheme: SUSHI retains interface unknowns where the diffusion tensor is discontinuous and imposes eliminable relations on other interfaces, combining hybrid and cell-centred treatments.With B denoting retained interfaces and H the other interior interfaces, the scheme has card(M) + card(H) equations and unknowns.
- Convergence and error estimation: The analysis proves convergence for general heterogeneous, anisotropic, possibly discontinuous tensors without solution regularity, and derives error estimates when the tensor and solution are regular enough.The discrete gradient and flux properties underpin these results.
3 Numerical results
The numerical experiments compare SUSHI variants across conforming, nonconforming, and heterogeneous meshes, including a tilted high-contrast barrier. Results show accurate fluxes and solutions, with method choice depending on mesh geometry and interface placement.
- Experimental setup: The experiments compare HFV, SUSHI-NP, and other choices of the interface set B on conforming, nonconforming, and heterogeneous problems.SUSHI-NP uses edges located on diffusion-tensor discontinuities, while HFV uses B = ∅.
- Experimental setup: The implementation assembles the linear system by looping over control volumes and their edges, with edge unknowns eliminated locally when required.The procedure applies to B = ∅, B ≠ ∅, and B = Eint; for HFV, cell unknowns may alternatively be eliminated.
- Tilted barrier test: In the tilted barrier test, the pure hybrid and composite schemes reproduce the analytical solution when discontinuity interfaces are excluded from B and each relevant cell remains within one subdomain.These conditions concern interfaces on ϕi(x, y) = 0 and the subdomain membership of control volumes adjacent to selected edges.
- Tilted barrier test: SUSHI on Mesh 1 is the most suitable method for the coupled tilted-barrier problem because Mesh 3 has control volumes that are too small.Mesh 3 still gives acceptable boundary-flux computations, but its geometry is unsuitable for the coupled problem.
- Tilted barrier test: Matching the hybrid interface set H to diffusion-tensor discontinuities is a satisfying choice, whereas B = ∅ can be preferable for numerical locking.The paper also notes that stabilization can preserve positivity on difficult meshes, generally with a larger L2 solution error.
4 Convergence of the scheme
The scheme’s fluxes satisfy continuity, coercivity, consistency, and symmetry, enabling convergence on general meshes without solution regularity assumptions. Under additional regularity, an isotropic-case error estimate is obtained.
- Bounded discrete solutions have weakly convergent discrete gradients, whose limit is the gradient of an H1_0(Ω) function.
- The constructed numerical fluxes form a continuous, coercive, consistent, and symmetric family.
- The discrete solution Π_Mu_D converges in L2(Ω) to the unique continuous solution, while ∇_Du_D converges in L2(Ω)^d to ∇u as h_D tends to zero.
- For Λ = Id and sufficiently regular solutions, the paper derives an error estimate, with an additional estimate for the specified flux construction.
- The error-estimate extension to general L∞ diffusion operators is unavailable because the continuous solution may have no more than H1 regularity.
5 Discrete functional analysis
The paper develops discrete functional-analytic tools for approximate solutions on general meshes, establishing embeddings, compactness, and convergence-supporting estimates.
- Discrete Sobolev embeddings: Discrete Sobolev inequalities are developed for mesh-based functions under mesh-ratio assumptions, including embeddings from discrete W 1,p norms into Lq spaces.For 1 ≤ p < ∞, some q > p depends only on p, while constants also depend on d, Ω, p, and the mesh-ratio parameter η.
- Compactness tools: The discrete compactness arguments use translation estimates and the Kolmogorov compactness theorem, with a BV-based proof available for the p = 1 case.For p > 1, the cited prior translation estimate requires orthogonality, whereas the present p = 1 proof handles general partitions.
- Compactness: For p = 1, bounded discrete W 1,1 norms yield relative compactness in L1 for families over general polyhedral partitions.The compactness also holds in L1(Rd) when the discrete functions are extended by zero outside Ω.
- Compactness: For 1 ≤ p < ∞, bounded discrete W 1,p norms yield relative compactness in Lp for mesh families satisfying the stated mesh-ratio condition.The result likewise applies in Lp(Rd) after zero extension outside Ω.
- Convergence support: The convergence framework produces a subsequence converging in Lp(Ω) to a limit function.This compactness result supplies the subsequence extraction needed in the convergence analysis.
6 Conclusion and perspectives
The paper presents a symmetric scheme for anisotropic heterogeneous diffusion on distorted nonconforming meshes and analyzes it through a discrete weak formulation. It proves convergence without solution regularity beyond the natural H1_0(Ω) setting, while numerical experiments report order 2 L2 convergence and an order 1 estimate for the Laplace operator.
- Conclusion and perspectives: The symmetric scheme handles anisotropic heterogeneous problems on distorted nonconforming meshes and can be viewed as a nonconforming finite element method.Its formulation is derived from a discrete weak formulation, although it stems from finite volume analysis.
- Conclusion and perspectives: Convergence of the discrete solution to the exact solution is shown without regularity assumptions beyond u ∈ H1_0(Ω).The result provides no convergence rate, but applies to the analyzed general setting.
- Conclusion and perspectives: An order 1 error estimate is established for the Laplace operator and is readily adaptable to regular piece-wise C1 isotropic diffusion operators.The paper contrasts this rate result with the general convergence theorem, which does not provide a rate.
- Conclusion and perspectives: The numerical results show good scheme performance, including order 2 convergence in the L2 norm of the solution.The conclusion also notes three-dimensional experiments on general grids performed for incompressible Navier–Stokes equations.
- Perspectives: The convergence analysis is stated to extend readily to nonlinear Leray–Lions operators, identified as future work.The extension is presented as a perspective rather than a result developed in this paper.