Reduced basis methods for low frequency electromagnetic problems

Abstract

In this thesis we introduce a new way for adaptively selecting snapshots to con- struct a reduced basis (RB) subspace for the numerical solution of parabolic differential equations. Our main insterest is in low frequency electromagnetic equations where the displacement currents can be neglected. Constructing the RB subspace by solving shifted stationary problems is a natural and often used attempt, based on the work of Grimme [7]. In [9] the identity of shifts and eigen- values of the reduced system was derived as a necessary optimality condition. Because the optimal spaces are not nested, Druskin, Lieberman and Zaslavsky proposed the usage of a nested sequence of spaces with adaptivly chosen shifts fitted to the eigenvalues in [6]. Using a modified version of the Kolmogorov Smirnow test statistic we derive an algorithm with a better fitting of the shifts to the eigenvalues than in [6]. We present tests on a 2D heat equation example and a 3D electromagnetic one and observe an improved convergence rate with our method.