An iterative spatial-stepping numerical method for linear elliptic PDEs using the Unified Transform

Abstract

A method for solving boundary value problems usually referred to as the unified transform or the Fokas method was introduced in the late nineties. A crucial role in this method is played by the so-called global relation, which characterizes the associated generalized Dirichlet-to-Neumann map, which can be used to determine the unknown boundary values in terms of the given boundary data. Then, the solution can be expressed via an integral formulated in the complex Fourier plane. This method can be considered as the spectral analogue of the classical Boundary Integral Method, which is formulated in the physical plane. Recently, it has been realized that the numerical implementation of the unified transform leads to a collocation-type method: this involves expanding the unknown boundary values in terms of appropriate basis functions and choosing a suitable set of complex collocation points. Herewith, an iterative spatial-stepping algorithm for computing the solution of a linear PDE in the interior of a convex polygon, is presented. The starting point of this algorithm is the analysis of the approximate global relation and its numerical solution. In more details, the interior of the polygon is partitioned into smaller concentric polygonal domains where the solution is computed recursively starting from the boundaries and proceeding towards the center. The solution of each inner polygon is computed by a spatial-stepping scheme using the Dirichlet and Neumann values of the respective outer polygon, which are computed via the approximate global relation.