Boundary Method of Weighted Residuals with Discontinuous Basis Functions for High-Accuracy Solving Linear Boundary Value Problems with Laplace and Poisson’s Equation

In the present paper the method of least squares with T-elements for solving linear boundary value problems with Laplace and Poisson’s equations is developed. In this approach it is offered to use discontinuous basis functions of a high-order approximation from special functional spaces, elaborated by the authors earlier. Advantage of the algorithm in comparison with Galerkin’s standard method is that, in the process of adaptive solving, it makes possible to condense economically a mesh and, moreover, to use different order of approximation of the solution on each cell of partition of calculated region. In contrast to Galerkin’s method with discontinuous basis functions, a penalty parameter here is not required, and the matrix of a discretized problem also is symmetric and positively definite. Examples of calculations by means of the schemes providing computer accuracy of the solution of boundary value problems for polynomials up to seventh order inclusive are given. In a three-dimensional case h − p-convergence of approximate solution to the exact one is shown.

boundary method of weighted residuals, discontinuous basis functions, T-elements, high accuracy, Laplace’s equation, Poisson’s equation