Galerkin-collocation approximation in time for the wave equation and its post-processing

Abstract

We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error estimates are proved for fully discrete approximations based on the Galerkin-collocation approach. Further, the concept of Galerkin-collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct connection between the two families by a computationally cheap post-processing is presented. The error estimates are illustrated by numerical experiments.

Variational Time Discretizations of Higher Order and Higher Regularity

“…Hence, all methods with even k share their strong A-stability with the dG method while methods with odd k are A-stable as the cGP method, cf. Remark 2.1 and [6]. ♣ Remark 5.4…”

“…where • denotes the Euclidean norm in R d . 1 presents the results for Q 6 0 -VTD 6 0 which is just dG (6) with numerical quadrature by the right-sided Gauss-Radau formula with 7 points. We show norms of the error between the solution u and the discrete solution U as well as the error between the solution u and the postprocessed discrete solution U in different norms.…”

“…Moreover, temporal derivatives of discrete solutions to affine linear systems with time-independent coefficients form also solutions of variational time discretization schemes. This relation was used to prove optimal error estimates for stabilized finite element methods for linear first-order partial differential equations [13] and for parabolic wave equations [6,8].…”

Higher order Galerkin-collocation time discretization with Nitsche's method for the Navier-Stokes equations

“…In the past, space-time finite element methods with continuous and discontinuous discretizations of the time and space variables have been studied strongly for the numerical simulation of incompressibe flow, wave propagation, transport phenomena or even problems of multi-physics; cf., e.g., [1,2,5,6,25,26,27,30,31,43,44,45]. Appreciable advantage of variational space-time discretizations is that they offer the potential to naturally construct higher order methods.…”

“…Recently, a modification of the standard continuous Galerkin-Petrov method (cGP) for the time discretization was introduced for wave problems (cf. [5,6,10]). The modification comes through imposing collocation conditions involving the discrete solution's derivatives at the discrete time nodes while on the other hand downsizing the test space of the discrete variational problem compared with the standard cGP approach.…”