Space–time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains

The spectral method is proposed for the partial integro-differential equations with a weakly singular kernel on irregular domains. The space discretization is based on the nodal spectral element method using the Lagrange polynomials basis associated with the Gauss–Lobatto–Legendre quadrature nodes. Also the model is discretized in time with the Legendre spectral Galerkin method. The discretization leads to conversion of the problem to a Sylvester matrix equation which can be solved efficiently by the QZ algorithm (Gardiner et al., 1992). The convergence of the method is proven by providing a priori L2L2-error estimate. Numerical results illustrate the efficiency and spectral accuracy of the proposed method.