TY - JOUR T1 - On a Galerkin Discretization of 4th order in Space and Time Applied to the Heat Equation AU - SHAFQAT HUSSAIN, FRIEDHELM SCHIEWECK, STEFAN TUREK, AND PETER ZAJAC JO - International Journal of Numerical Analysis Modeling Series B VL - 4 SP - 353 EP - 371 PY - 2013 DA - 2013/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnamb/262.html KW - continuous Galerkin-Petrov method KW - nonconforming FEM KW - heat equation KW - multigrid method AB - We present a new time discretization scheme based on the continuous Galerkin Petrov method of polynomial order 3 (cGP(3)-method) which is combined with a reduced numerical time integration (3-point Gauß-Lobatto formula). The solution of the new approach can be computed from the solution of the lower order cGP(2)-method, which requires to solve a coupled 2 × 2 block system on each time interval, followed by a simple post-processing step, such that we get the higher accuracy of 4th order in time in the standard L²-norm with nearly the cost of the cGP(2)-method. Moreover, the difference of both solutions can be used as an indicator for the approximation error in time. For the approximation in space we use the nonparametric \tilde{Q}_3-element which belongs to a family of recently derived higher order nonconforming finite element spaces and leads to an approximation error in space of order 4, too, in the L²-norm. The expected optimal accuracy of the full discretization error in the L²-norm of 4th order in space and time is confirmed by several numerical tests. We discuss implementation aspects of the time discretization as well as efficient multigrid methods for solving the resulting block systems which lead to convergence rates being almost independent of the mesh size and the time step. In our numerical experiments we compare different higher order spatial and temporal discretization approaches with respect to accuracy and computational cost.