A nonconforming combined method for solving Laplace's boundary value problems with singularities
For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used. In this paper, we will develop such an approach by using a new coupling strategy, which is described as follows: If \(L+1=O(| \ell n h|)\), the average errors of numerical solutions and their generalized derivatives are still O(h), where h is the maximal boundary length of quasiuniform triangular elements in the finite element method, and \(L+1\) is the total number of singular admissible functions in the Ritz-Galerkin method. This coupling relation, \(L+1=O(| \ell n h|)\), is significant because only a few singular functions are required for a good approximation of solutions.
scientific article; zbMATH DE number 3940660
scientific article