Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

Abstract

We consider a second order differential operator A(x) = − d i,j=1, on a bounded domain D with Dirichlet boundary conditions on ∂D, under mild assumptions on the coefficients of the diffusion tensor aij . The object is to construct monotone numerical schemes to approximate the solution to the problem A(x) u(x) = µ(x), x ∈ D, where µ is a positive Radon measure. We start by briefly mentioning questions of existence and uniqueness, introducing function spaces needed to prove convergence results. Then, we define non-standard stencils on grid-knots that lead to extended discretization schemes by matrices possesing compartmental structure. We proceed to discretization of elliptic operators, starting with constant diffusion tensor and ending with operators in divergence form. Finally, we discuss W 1 2 -convergence in detail, and mention convergence in C and L1 spaces. We conclude by a numerical example illustarting the schemes and convergence results.

Monotone schemes for a class of nonlinear elliptic and parabolic problems

“…The method is described in [7,8]. In addition, this result satisfies our needs for theoretical grounds of nonlinear BVPs to be studied in Sections 5, 6.…”

“…Their structure follows from the corresponding discretized forms [7,8]. The diagonal entries are the negative sums of off-diagonal entries.…”

“…Discretizations A n of (8) and (9) are obtained by applying the variational calculus to certain bilinear forms (v n , u n ) → a n (v n , u n ) which discretize the original bilinear form a(1; ·, ·) [7,8]. These forms can be expressed as the sums of backward/forward finite difference formulae with various step sizes.…”

“…Construction of discretizations A n with the compartmental structure for d ≥ 3 is based on the reduction to two-dimensional problems, and one can prove [7,8]:…”

“…They have the structure as demanded by Theorem 3.2. For examples with a diffusion with jumps, for the linear problem, see [6][7][8].…”