Numerical solution of an elliptic problem with several interfaces

An algorithm of the high-accuracy numerical solution of the second order elliptic equation with several interfaces including intersecting and non-convex ones is developed. To approximate the interface problem in the neighbourhood of the discontinuity lines irregular cells (i-cells) which are cut off by the discontinuity lines from the regular cells of the rectangular grid and the “outsidethe-contour” parts of the cells are used. To construct an approximate solution, it is proposed: 1) to write out the additionally matching conditions in i-cells on interfaces increasing the number of matching cells; 2) to reduce the common part of the discontinuity line enclosed between neighboring cells and used for setting conditions. To solve the Dirichlet boundary value problem the hp-version of the least-squares collocation method (hp-LSCM) is implemented in combination with modern algorithms for accelerating the iterative process: preconditioning, parallelization of the computational program using OpenMP, Krylov subspaces; multigrid method. The convergence of the hp-LSCM and the conditionality of the arising overdetermined systems of linear algebraic equations (SLAE) are investigated in solving various test problems. The results obtained by the LSCM and other authors using the method MIB (matched interface and boundary) are compared.