A POD based extrapolation DG time stepping space-time FE method for parabolic problems

Abstract

Based on the proper orthogonal decomposition technique, a new reduced-order discontinuous time stepping space-time finite element extrapolation iterative scheme is constructed to solve two-dimensional parabolic problems, which includes very few degrees of freedom but holds sufficiently high accuracy. A reduced space-time projection operator is defined, and its related error estimations are discussed. Then, the error estimate between the classical scheme solution and the reduced-basis scheme solution, and the a priori error estimate for the reduced-basis scheme are derived, respectively. These error estimates are analyzed by using the technique of combining Radau quadrature rule with finite element method, without considering any dual problems. The algorithm implementation of the reduced-order extrapolation iterative scheme is also provided. Finally, a numerical example is presented. The numerical results are consistent with theoretical ones. Moreover, it is shown that the new reduced-order extrapolation iterative scheme, by comparing to the standard approach, can reduce CPU time without loss of accuracy. Therefore, it is feasible and efficient for solving parabolic problems.