hp-version Galerkin time-stepping with Laguerre spectral methods for parabolic equations on semi-unbounded domains

Abstract

We propose fully discrete numerical methods, high-order in both space and time, for solving parabolic equations on semi-unbounded domains. The spatial discretization uses a spectral method based on Laguerre functions, while the temporal discretization employs two distinct hp-version Galerkin time-stepping schemes: the continuous Petrov-Galerkin and the discontinuous Galerkin methods. A priori error estimates are derived in the \(L^2(0,T;H^1(\mathbb {R}_{+}))\)-norm, with all constants explicitly independent of the temporal and spatial discretization parameters. In the presence of singularities at \(t = 0\), exponential rates of convergence in time are obtained by using geometrically refined time meshes together with approximation degrees that increase linearly. Numerical experiments are conducted to verify the theoretical findings.