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.