Boundary least squares method with three-dimensional harmonic basis of higher order for solving linear div-curl systems with Dirichlet conditions

Abstract

Abstract Solving linear divergence-curl system with Dirichlet conditions is reduced to finding an unknown vector function in the space of piecewise-polynomial gradients of harmonic functions. In this approach one can use the boundary least squares method with a harmonic basis of a high order of approximation formulated by the authors previously. The justification of this method is given. The properties of the bilinear form and approximating properties of the basis are investigated. Convergence of approximate solutions is proved. A numerical example with estimates of experimental orders of convergence in $\begin{array}{} {\bf V}_h^p \end{array}$-norm for different parameters h, p (p ⩽ 10) is presented. The method does not require specification of penalty weight function.