Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation

Abstract

We discuss numerical properties of continuous Galerkin-Petrov and discontinuous Galerkin time discretizations applied to the heat equation as a prototypical example for scalar parabolic partial differential equations. For the space discretization, we use biquadratic quadrilateral finite elements on general two-dimensional meshes. We discuss implementation aspects of the time discretization as well as efficient methods for solving the resulting block systems. Here, we compare a preconditioned BiCGStab solver as a Krylov space method with an adapted geometrical multigrid solver. Only the convergence of the multigrid method is almost independent of the mesh size and the time step leading to an efficient solution process. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs.

A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

“…Moreover, the cGP-method is Astable [2] and the dG-method is L-stable [3] for each polynomial degree k in time. In our recent paper [1], we described both methods in detail for the heat equation and proposed an efficient multigrid method for solving the according block systems in each time step. For example, we have demonstrated by means of numerical test problems that the cGP(2)-method is of third order accurate in each time point and even of fourth order in the endpoints t n of the time *Address correspondence to this author at the Institut für Angewandte Mathematik, TU Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Germany; Tel: +49-(0) 231-755-7216; Fax: +49-(0) 231-755-5933; E-mail: shafqat.hussain@math.uni-dortmund.de intervals whereas the dG(1)-method has the order two in each time point and order three in the discrete time points t n .…”

“…The method was analyzed for the linear case in an abstract Hilbert space and for the non-linear case in the Euclidean space. In [1] the method was renamed into its final name ''continuous Galerkin Petrov'' method since the ansatz space is continuous in time contrary to the well-known ''discontinuous Galerkin'' method [3]. The dG-method in time has already a long tradition in literature, see e.g.…”

“…Therein, for an abstract model problem, which includes again the heat equation, optimal error estimates and superconvergence results at the discrete time points are proven. In [1] we have compared for the heat equation, the cGP( k )-method, where the time polynomial ansatz is of degree k , with the dG( k 1 )-method where both ansatzand test-space are time polynomial of degree k 1 . Since both methods require to solve a linear k k -block system in each time step, they have comparable computational costs concerning computing time and memory requirements.…”

“…In this paper, we extend our work on the heat equation in [1] to the nonstationary Stokes equations. The new difficulty is now that we have to treat a saddle point problem arising from the mixed formulation with velocity and pressure.…”

“…

Abstract:In this note, we extend our recent work for the heat equation in [1] and describe and compare by means of numerical experiments the continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization applied to the nonstationary Stokes equations in the two-dimensional case. For the space discretization, we use the well-known LBB-stable quadrilateral finite element which consists of conforming biquadratic elements for the velocity and discontinuous linear elements for the pressure.

…”