High accuracy numerical solution of elliptic equations with discontinuous coefficients

We develop an approach to constructing a new high-accuracy hp-version of the least-squares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations with a coefficient discontinuity on lines of different shapes in a problem solution domain. In order to approximate the equation and the conditions on the discontinuity of its coefficient, it is proposed to use the external parts and irregular cells (i-cells) of the computational grid which are cut off by the line of discontinuity from regular rectangular cells. The proposed approach allows to obtain solutions with a high order of convergence and high accuracy by grid refining and/or increasing the degree of the approximating polynomials both in the case of the Dirichlet conditions on the boundary of the domain and in the case of the presence of Neumann conditions on a large part of the boundary. Also, we consider the case of the problem with a discontinuity of the second derivatives of the desired solution in addition to the coefficient discontinuity at the corner points of the domain. We simulate the heat transfer process in the domain where particles of the medium move in a plane-parallel manner with a phase transition and heat release at the front of the discontinuity line. An effective combination of the LSC method with various methods of accelerating the iterative process is demonstrated: the acceleration algorithm based on Krylov subspaces; the operation of prolongation along the ascending branch of the V-cycle on a multigrid complex; parallelization. The results are compared with those of other authors on solving the considered problems.