An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem

Abstract

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p≥3, we derive the a priori error estimates of the approximate eigenfunction in the broken H1 norm and L2 norm which are optimal in h and suboptimal in p. When p=2, the approximate eigenfunctions converge but with only suboptimal convergence rate. When p≥2 and the eigenfunction u∈Hs(Ω)(s≥p+1), we prove that the convergence rate of approximate eigenvalues reaches 2p−2 in h and p−(2s−7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues.