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

Author(s):  
Jake Harmon ◽  
Jeremiah Corrado ◽  
Branislav Notaros

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.

2021 ◽  
Author(s):  
Jake Harmon ◽  
Jeremiah Corrado ◽  
Branislav Notaros

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.


2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva

AbstractIn 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.


2019 ◽  
Vol 144 (2) ◽  
pp. 323-346 ◽  
Author(s):  
M. Feischl ◽  
Ch. Schwab

AbstractFor functions $$u\in H^1(\Omega )$$u∈H1(Ω) in a bounded polytope $$\Omega \subset {\mathbb {R}}^d$$Ω⊂Rd$$d=1,2,3$$d=1,2,3 with plane sides for $$d=2,3$$d=2,3 which are Gevrey regular in $$\overline{\Omega }\backslash {\mathscr {S}}$$Ω¯\S with point singularities concentrated at a set $${\mathscr {S}}\subset \overline{\Omega }$$S⊂Ω¯ consisting of a finite number of points in $$\overline{\Omega }$$Ω¯, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in $$\Omega $$Ω. The simplicial meshes are geometrically refined towards $${\mathscr {S}}$$S but are otherwise unstructured.


2001 ◽  
Vol 11 (02) ◽  
pp. 301-337 ◽  
Author(s):  
K. GERDES ◽  
J. M. MELENK ◽  
C. SCHWAB ◽  
D. SCHÖTZAU

The Streamline Diffusion Finite Element Method (SDFEM) for a two-dimensional convection–diffusion problem is analyzed in the context of the hp-version of the Finite Element Method (FEM). It is proved that the appropriate choice of the SDFEM parameters leads to stable methods on the class of "boundary layer meshes", which may contain anisotropic needle elements of arbitrarily high aspect ratio. Consistency results show that the use of such meshes can resolve layer components present in the solutions at robust exponential rates of convergence. We confirm these theoretical results in a series of numerical examples.


1994 ◽  
Vol 61 (4) ◽  
pp. 919-922 ◽  
Author(s):  
Taein Yeo ◽  
J. R. Barber

When heat is conducted across an interface between two dissimilar materials, theimoelastic distortion affects the contact pressure distribution. The existence of a pressure-sensitive thermal contact resistance at the interface can cause such systems to be unstable in the steady-state. Stability analysis for thermoelastic contact has been conducted by linear perturbation methods for one-dimensional and simple two-dimensional geometries, but analytical solutions become very complicated for finite geometries. A method is therefore proposed in which the finite element method is used to reduce the stability problem to an eigenvalue problem. The linearity of the underlying perturbation problem enables us to conclude that solutions can be obtained in separated-variable form with exponential variation in time. This factor can therefore be removed from the governing equations and the finite element method is used to obtain a time-independent set of homogeneous equations in which the exponential growth rate appears as a linear parameter. We therefore obtain a linear eigenvalue problem and stability of the system requires that all the resulting eigenvalues should have negative real part. The method is discussed in application to the simple one-dimensional system of two contacting rods. The results show good agreement with previous analytical investigations and give additional information about the migration of eigenvalues in the complex plane as the steady-state heat flux is varied.


Sign in / Sign up

Export Citation Format

Share Document