On the Combination of the Laplace Transform and the Integral Equation Method to the 3D Parabolic Initial Boundary Value Problem

We consider a two-step numerical approach for solving parabolic initial boundary value problems in the 3D simply connected smooth regions. The method uses the Laplace transform in time, reducing the problem to a set of independent stationary boundary value problems for the Helmholtz equation with complex parameters. The inverse Laplace transform is computed using sinc quadrature along a suitably chosen contour in the complex plane. We showed that due to a symmetry of the quadrature nodes, the number of stationary problems can be decreased by almost a factor of 2. The influence of the integration contour parameters on the approximation error is also researched. Stationary problems are numerically solved using boundary integral equation approach applying Nystr\"om method, based on the quadratures for smooth surface integrals. Numerical experiments support the expectations.