Abstract—

A method is presented for solving interior boundary-value problems for second-order elliptic equations by transition to ray variables. The domain is divided into cells within which the coefficients and sources have the smoothness and continuity properties necessary for the existence of a regular classical solution in the cell. The finite discontinuities of the coefficients (if any) are located on the cell boundaries. The regular solution in the cell is sought in the form of a superposition of the contributions made by volume and boundary sources placed on the rays arriving at the given point from the cell boundaries. Next, a finite analytic scheme for the numerical solution of the boundary value problem in a domain with discontinuous coefficients and sources is constructed by matching the regular solutions emerging from cells at the cell boundaries. The scheme exhibits no hard dependence of the accuracy of approximation on the sizes and shape of the cells, which is inherent in finite-difference schemes.

中文翻译：

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关于二阶线性椭圆方程的求解

摘要—

提出了一种通过过渡到射线变量来求解二阶椭圆方程内部边值问题的方法。该域被划分为多个单元，其中的系数和源具有单元中存在常规经典解所需的平滑度和连续性。系数的有限不连续点（如果有）位于单元边界上。在单元中寻求规则解的形式是将体积和边界源的贡献叠加在一起，该体积和边界源放置在从单元边界到达给定点的射线上。下一个，通过匹配从单元边界处的单元格出现的正则解，构造了一个具有不连续系数和源的域中边值问题的数值解的有限解析方案。该方案没有表现出近似精度对像元大小和形状的严格依赖，这是有限差分方案中固有的。