Space-time hp-approximation of parabolic equations
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2017-10
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Report
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Abstract
A new space-time finite elements methods (FEM) approximation for the solution of parabolic partial differential equations (PDE) is introduced. Considering a mesh-dependent norm, it is first shown that the discrete bilinear form is coercive and continuous, yielding existence and uniqueness of the associated solution. In a second step, error estimates in this 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.
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2017-50
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Seminar for Applied Mathematics, ETH Zurich
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03435 - Schwab, Christoph / Schwab, Christoph
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