A Refinement-by-Superposition hp-Method for H(curl)- and H(div)-Conforming Discretizations

We present an application of refinement-by-superposition (RBS) <i>hp</i>-refinement in computational electromagnetics (CEM), which permits exponential rates of convergence. In contrast to dominant approaches to <i>hp</i>-refinement for continuous Galerkin methods, which rely on constrained-nodes, the multi-level strategy presented drastically reduces the implementation complexity. Through the RBS methodology, enforcement of continuity occurs by construction, enabling arbitrary levels of refinement with ease and without the practical (but not theoretical) limitations of constrained-node refinement. We outline the construction of the RBS <i>hp</i>-method for refinement with <i>H</i>(curl)- and <i>H</i>(div)-conforming finite cells. Numerical simulations for the 2-D finite element method (FEM) solution of the Maxwell eigenvalue problem demonstrate the effectiveness of RBS <i>hp</i>-refinement. An additional goal of this work, we aim to promote the use of mixed-order (low- and high-order) elements in practical CEM applications.

A Refinement-by-Superposition hp-Method for H(curl)- and H(div)-Conforming Discretizations

We present an application of refinement-by-superposition (RBS) <i>hp</i>-refinement in computational electromagnetics (CEM), which permits exponential rates of convergence. In contrast to dominant approaches to <i>hp</i>-refinement for continuous Galerkin methods, which rely on constrained-nodes, the multi-level strategy presented drastically reduces the implementation complexity. Through the RBS methodology, enforcement of continuity occurs by construction, enabling arbitrary levels of refinement with ease and without the practical (but not theoretical) limitations of constrained-node refinement. We outline the construction of the RBS <i>hp</i>-method for refinement with <i>H</i>(curl)- and <i>H</i>(div)-conforming finite cells. Numerical simulations for the 2-D finite element method (FEM) solution of the Maxwell eigenvalue problem demonstrate the effectiveness of RBS <i>hp</i>-refinement. An additional goal of this work, we aim to promote the use of mixed-order (low- and high-order) elements in practical CEM applications.

Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.