Convergence of Finite-Difference Schemes for Elliptic Equations with Variable Coefficients

Abstract

Abstract. First boundary value problem for elliptic equation with youngest coefficient containing Dirac distribution concentrated on a smooth curve is considered. For this problem a finite difference scheme on a special quasiregular grid is constructed. The finite difference scheme converges in discrete W 1 2 norm with the rate O(h 3/2 ). Convergence rate is compatible with the smoothness of input data.

Supraconvergence and Supercloseness of a Scheme for Elliptic Equations on Nonuniform Grids

“…Schemes for less regular coefficients (on uniform grids) are also known [11,12,15,20,28], which are based on earlier work by Samarskij [27]. We also impose the general assumption that the homogeneous problem (2.1), i.e., with g = 0 and = 0, has only the solution u = 0.…”

“…So it appears natural, as our results show, that the H 1 error estimates obtained for the FDM are closely related to supercloseness of the FEM. But the literature gives the impression that there exist the two separated communities of the FEM and FDM people (see [12,13,15,28] and the overview in [11] for the latter) and the relation of those results has not been considered in that respect.…”

Convergence Of a Finite Difference Scheme for the Third Boundary-Value Problem for an Elliptic Equation with Variable Coefficients

“…Let us consider the sum where the mesh function v is extended outsideω by (9). Using the periodicity and orthogonality of sines, from (9) follows…”

“…The term η ij can be estimated in the same manner as in the case of the Dirichlet boundary-value problem (see [9], [7]):…”

Convergence rates of finite difference schemes for the linear advection and wave equation with rough coefficient

“…It is shown that the approximations converge at rate ∆x λ−2 in the energy norm (a discrete version of the H 1 -norm). In [10], this result is extended to coefficients a ij (x) ∈ W λ−1,2 (D), a(x) ∈ W λ−2,2 (D). Another related work is the article [13] by V. Jovanović and C. Rohde where they establish error estimates for finite volume approximations of linear hyperbolic systems in multiple space dimensions for initial data with low regularity, however under the assumption that the coefficients are smooth.…”