Spectral/hp Hull: A Degree of Freedom Reducing Discontinuous Spectral Element Method with an Application in Compressible Fluid Flow

摘要

A concept for reducing the degrees of freedom (DOF) of discontinuous finite element methods is proposed using a systematic hp-process. The elements are first agglomerated (h-coarsening) to form convex/concave hulls and then the polynomial degree of the hull basis is increased (p-refinement). Compared to the conventional discontinuous FEM, this mechanism yields more accurate solutions with smaller DOF. This methodology is evaluated in the standard Discontinuous Galerkin (DG) and Discontinuous Least-Squares (DLS) spectral hull formulations. The classical DG and DLS elements require additional (duplicate) surface nodes and hence are known to be less efficient than Continuous Galerkin (CG) formulations. However, in this paper, it will be shown that this is not true when hull basis are used. Therefore, this concept may eliminate the only disadvantage of popular discontinuous finite element methods. The accuracy and efficiency of the proposed method is demonstrated using the linear acoustics equations and on a two-dimensional compressible vortex shedding problem.