Linear Complexity Solution of Parabolic Integro-differential Equations

The numerical solution of parabolic problems with a pseudo-differential operator by wavelet discretization in space and hp discontinuous Galerkin time stepping is analyzed. It is proved that an approximation for u(T) can be obtained in N points with accuracy for any integer p ≥ 1 in work and memory which grows logarithmically-linear in N.Supported in part IHP Network Breaking Complexity of the EC (contract number HPRN-CT-2002-00286) with support by the Swiss Federal Office for Science and Education under grant No. BBW 02.0418.Funded by the Swiss National Science Foundation (Grant PBEZ2-102321).     

Two-Grid hp-Version DGFEMs for Strongly Monotone Second-Order Quasilinear Elliptic PDEs

In this article we develop the a priori error analysis of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of strongly monotone second-order quasilinear partial differential equations. In this setting, the fully nonlinear problem is first approximated on a coarse finite element space V(𝒯_H, P ). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearize the underlying discretization on the finer space V(𝒯_h, p ); thereby, only a linear system of equations is solved on the richer space V(𝒯_h, p ). Numerical experiments confirming the theoretical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)