High-order accurate difference schemes for elliptic equations in a domain with a curvilinear boundary

A. V. Shapeev, V. P. Shapeev

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


A method is suggested for constructing high-order accurate difference schemes for solving boundary value problems for elliptic equations with variable coefficients in a domain with a curvilinear boundary. A scheme is constructed by the method of indeterminate coefficients in the form of a linear combination of the values of the desired solution at the stencil points. Regular stencils constructed from squares or regular triangles are used inside the domain; irregular stencils, on the domain boundary. Procedures for analytical and numerical construction of schemes are implemented in a software code. In the latter case, for a given equation and boundary conditions, the coefficients of difference equations are found numerically at each point of the computational domain in the course of a solution. The order of accuracy of the constructed schemes is not greater than four in the former case, and schemes of up to the 16th order are constructed in the latter case. Examples of new high-order accurate schemes are given. The convergence of difference solutions on a sequence of meshes is illustrated by numerous numerical experiments. Starting from a certain mesh size, the order of convergence corresponds to the approximation order in many cases. The runtime required to solve a problem increases with the order of the scheme, but the solution accuracy increases faster than the runtime.

Original languageEnglish
Pages (from-to)213-221
Number of pages9
JournalComputational Mathematics and Mathematical Physics
Issue number2
Publication statusPublished - 2000
Externally publishedYes


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