Solving Eigenvalue Problems in a Discontinuous Approximation Space by Patch Reconstruction

Abstract

We adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in two dimensions and three dimensions. It is a direct extension of the method recently proposed to solve corresponding boundary value problems, and the optimal error estimates of the approximation to eigenfunctions and eigenvalues are instant consequences from existing results. The method enjoys the advantage that it uses only one degree of freedom on each element to achieve very high order accuracy, which is highly preferred for eigenvalue problems as implied by Zhang's recent study [J. Sci. Comput., 65(2015), pp. 455--466]. By numerical results, we illustrate that higher order methods can provide much more reliable eigenvalues. To justify that our method is the right one for eigenvalue problems, we show that the patch reconstructed approximation space attains the same accuracy with much less degrees of freedom than classical discontinuous Galerkin methods. With the increasing of the polynomial order, our method can even achieve better performance than conforming finite element methods; such methods are traditionally the methods of choice to solve problems with high regularities.