Construction and Convergence Study of Schemes Preserving the Elliptic Local Maximum Principle

Abstract

Abstract. We present a method to approximate (in any space dimension) diffusion equations with schemes having a specific structure; this structure ensures that the discrete local maximum and minimum principles are respected, and that no spurious oscillations appear in the solutions. When applied in a transient setting on models of concentration equations, it guaranties in particular that the approximate solutions stay between the physical bounds. We make a theoretical study of the constructed schemes, proving under a coercivity assumption that their solutions converge to the solution of the PDE. Several numerical results are also provided; they help us understand how the parameters of the method should be chosen. These results also show the practical efficiency of the method, even when applied to complex models.

Finite volume schemes for diffusion equations: Introduction to and review of modern methods

“…23 have been used to construct MMP schemes on triangular meshes, 104,105 construction then generalized to generic 2D or 3D meshes in Ref. 58.…”

“…However, under some coercivity assumptions (which seem satisfied in numerical tests), a rigorous proof of convergence is given in Ref. 58 without regularity assumptions on the data, drawing on the fact that the global flux is a convex combination of linear fluxes and adapting the analysis technique developed in Ref. 10.…”

Existence of solution of a finite volume scheme preserving maximum principle for diffusion equations

“…The combination of our scheme with the method in can also lead to new finite volume schemes satisfying the discrete maximum principle. We should point out that, the first step of our construction does not exist in and , the second step is similar but not the same as that in the above two references.…”

“…The main contribution of is that it constructs a nonlinear scheme that provides an approximation of the solution at the centers of the cells (see Section 6.2 in ). In , a finite volume scheme is constructed to satisfy the discrete maximum principle for diffusion equations with tensor coefficients on polygonal meshes. The existence of the solution and the convergence properties are proved both in and .…”

A monotone nonlinear finite volume method for approximating diffusion operators on general meshes

“… The first family uses a projection of the initial mesh on an underlying Voronoi mesh; see . The second family proposes to add a nonlinear stabilization term to the piecewise linear Galerkin approximation of the diffusion operator; see . The third family that dates back to yields monotone finite volume methods based on a nonlinear two‐points flux approximation; see . In what follows, this type of method will be named nonlinear monotone two‐point finite volumes (NLMTPFV). The fourth family that traces back to yields nonlinear multi‐point flux approximation finite volume methods, which satisfy a discrete maximum principle; see and . The fifth family proposed in allows any cell‐centered method to be the basis for devising a new nonlinear multi‐point flux approximation finite volume method that satisfies the discrete maximum principle (see also and ). …”