Space–time hp-approximation of parabolic equations

A new space–time finite element method for the solution of parabolic partial differential equations is introduced. In a mesh and degree-dependent norm, it is first shown that the discrete bilinear form for the space–time problem is both coercive and continuous, yielding existence and uniqueness of the associated discrete solution. In a second step, error estimates in this mesh-dependent norm are derived. In particular, we show that combining low-order elements for the space variable together with an hp-approximation of the problem with respect to the temporal variable allows us to decrease the optimal convergence rates for the approximation of elliptic problems only by a logarithmic factor. For simultaneous space–time hp-discretization in both, the spatial as well as the temporal variable, overall exponential convergence in mesh-degree dependent norms on the space–time cylinder is proved, under analytic regularity assumptions on the solution with respect to the spatial variable. Numerical results for linear model problems confirming exponential convergence are presented.