On Local Artificial Boundary Conditions for the Diffusion Equation in Case of 2d Convex Computational Domain

We construct artificial boundary conditions (ABCs) for solving the diffusion equation in a 2D convex computational domain with a piecewise smooth artificial boundary. We start from the domain decomposition, representing the plane R R as a union of an interior (computational) and exterior domains with an overlap. Then we consider the exterior problem: applying the Laplace transform in time, as well as using the operator splitting by coordinates together with the spline interpolation, we obtain an infinite family of functions that approximate the exterior solution with higher and higher accuracy. These functions are used to impose local boundary conditions for the interior problem. Unlike the standard domain decomposition procedure for joining the solutions on the boundaries of the neighbouring domains – the Schwarz alternating method – we construct the ABCs in one step, without iterations. We also give estimates for the errors produced by the ABCs on the artificial boundary, and show that the resulting boundary value problems are well-posed. Numerical results are presented