Optimal trace inequality constants for interior penalty discontinuous Galerkin discretisations of elliptic operators using arbitrary elements with non-constant Jacobians

• Published in: Journal of computational physics 2017-12, Vol.350, p.847-870
• Main Authors: Owens, A.R., Kópházi, J., Eaton, M.D.
• Format: Article
• Language: English
• Subjects: Approximation; Coercivity; Computational physics; Conditioning; Constants; Diffusion; Diffusion synthetic acceleration; Discontinuous Galerkin; Eigenvalues; Finite element analysis; Finite element method; Galerkin method; Interior penalty; Isogeometric analysis; Jacobians; Linear systems; Mathematical models; Operators; Parameters; Physics; Splines; Stiffness matrix; Trace inverse inequality
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2017.09.020

•Numerical method for calculating optimal trace inequality constants for application to interior penalty methods presented.•Can be applied to any elements for which stiffness matrices can be calculated, including those with a non-constant Jacobian.•The new method provides sharper bounds than those from the literature when the mapping from the reference element is linear.•Results in a linear system that can be solved 11% faster than methods from the literature for an MMS test case. In this paper, a new method to numerically calculate the trace inequality constants, which arise in the calculation of penalty parameters for interior penalty discretisations of elliptic operators, is presented. These constants are provably optimal for the inequality of interest. As their calculation is based on the solution of a generalised eigenvalue problem involving the volumetric and face stiffness matrices, the method is applicable to any element type for which these matrices can be calculated, including standard finite elements and the non-uniform rational B-splines of isogeometric analysis. In particular, the presented method does not require the Jacobian of the element to be constant, and so can be applied to a much wider variety of element shapes than are currently available in the literature. Numerical results are presented for a variety of finite element and isogeometric cases. When the Jacobian is constant, it is demonstrated that the new method produces lower penalty parameters than existing methods in the literature in all cases, which translates directly into savings in the solution time of the resulting linear system. When the Jacobian is not constant, it is shown that the naive application of existing approaches can result in penalty parameters that do not guarantee coercivity of the bilinear form, and by extension, the stability of the solution. The method of manufactured solutions is applied to a model reaction-diffusion equation with a range of parameters, and it is found that using penalty parameters based on the new trace inequality constants result in better conditioned linear systems, which can be solved approximately 11% faster than those produced by the methods from the literature.