Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the hp-version

15 May 2014
14:00
Andrea Moiola
Abstract
<p><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Computer simulation of the propagation and interaction of linear waves</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">is a core task in computational science and engineering.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">The finite element method represents one of the most common</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">discretisation techniques for Helmholtz and Maxwell's equations, which</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">model time-harmonic acoustic and electromagnetic wave scattering.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">At medium and high frequencies, resolution requirements and the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">so-called pollution effect entail an excessive computational effort</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and prevent standard finite element schemes from an effective use.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">The wave-based Trefftz methods offer a possible way to deal with this</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">problem: trial and test functions are special solutions of the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">underlying PDE inside each element, thus the information about the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">frequency is directly incorporated in the discrete spaces.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">This talk is concerned with a family of those methods: the so-called</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Trefftz-discontinuous Galerkin (TDG) methods, which include the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">well-known ultraweak variational formulation (UWVF).</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">We derive a general formulation of the TDG method for Helmholtz</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">impedance boundary value problems and we discuss its well-posedness</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and quasi-optimality.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">A complete theory for the (a priori) h- and p-convergence for plane</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and circular/spherical wave finite element spaces has been developed,</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">relying on new best approximation estimates for the considered</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">discrete spaces.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">In the two-dimensional case, on meshes with very general element</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">shapes geometrically graded towards domain corners, we prove</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">exponential convergence of the discrete solution in terms of number of</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">unknowns.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Switzerland) and Ilaria Perugia (Vienna, Austria).</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /></p>
  • Computational Mathematics and Applications Seminar