-error analysis of a continuous Petrov–Galerkin time-stepping scheme for linear parabolic equations

We present a fully discrete hp-version numerical method for second-order linear parabolic equations, combining a continuous Petrov–Galerkin (CPG) time-stepping scheme with a standard continuous Galerkin (CG) finite element method in space. Our analysis provides a priori error estimates in the L2(L2)- and L∞(L2)-norms, where all constants are fully robust – independent of temporal and spatial discretization parameters. For solutions with initial singularities at t = 0, exponential temporal convergence is achieved through geometrically graded time meshes and linearly increasing polynomial degrees. Numerical experiments confirm the theoretical convergence rates.