scholarly journals Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

2013 ◽  
Vol 16 (6) ◽  
pp. 283-302 ◽  
Author(s):  
Peter Gerds ◽  
Lars Grasedyck
Keyword(s):  
Eigenvalue Problem ◽  
Hierarchical Matrices ◽  
Elliptic Pde ◽  
Multi Level

scholarly journals Accidental Degeneracy of an Elliptic Differential Operator: A Clarification in Terms of Ladder Operators

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3005
Author(s):  
Roberto De Marchis ◽  
Arsen Palestini ◽  
Stefano Patrì
Keyword(s):  
Differential Operator ◽  
Eigenvalue Problem ◽  
Second Order ◽  
Rotational Invariance ◽  
Polar Coordinates ◽  
Elliptic Pde ◽  
Elliptic Differential Operator ◽  
Ladder Operators ◽  
Accidental Degeneracy

We consider the linear, second-order elliptic, Schrödinger-type differential operator L:=−12∇2+r22. Because of its rotational invariance, that is it does not change under SO(3) transformations, the eigenvalue problem −12∇2+r22f(x,y,z)=λf(x,y,z) can be studied more conveniently in spherical polar coordinates. It is already known that the eigenfunctions of the problem depend on three parameters. The so-called accidental degeneracy of L occurs when the eigenvalues of the problem depend on one of such parameters only. We exploited ladder operators to reformulate accidental degeneracy, so as to provide a new way to describe degeneracy in elliptic PDE problems.

scholarly journals A multi-level mixed element scheme of the two-dimensional Helmholtz transmission eigenvalue problem

2018 ◽  
Vol 40 (1) ◽  
pp. 686-707 ◽  
Author(s):  
Yingxia Xi ◽  
Xia Ji ◽  
Shuo Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Computational Cost ◽  
Optimal Convergence Rate ◽  
Element Scheme ◽  
Polygonal Domains ◽  
Transmission Eigenvalue ◽  
Transmission Eigenvalue Problem ◽  
Rectangular Grids ◽  
Multi Level ◽  
Mixed Element

Abstract In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangular grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal computational cost can be obtained with the scheme.

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

2021 ◽  
Author(s):  
Jake Harmon ◽  
Jeremiah Corrado ◽  
Branislav Notaros
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Rates Of Convergence ◽  
Galerkin Methods ◽  
Mixed Order ◽  
Continuous Galerkin ◽  
Multi Level ◽  
Continuous Galerkin Methods ◽  
Exponential Rates

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.

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

2021 ◽  
Author(s):  
Jake Harmon ◽  
Jeremiah Corrado ◽  
Branislav Notaros
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Rates Of Convergence ◽  
Galerkin Methods ◽  
Mixed Order ◽  
Continuous Galerkin ◽  
Multi Level ◽  
Continuous Galerkin Methods ◽  
Exponential Rates

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.

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