scholarly journals EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

2005 ◽  
Vol 15 (04) ◽  
pp. 575-622 ◽  
Author(s):  
MARTIN COSTABEL ◽  
MONIQUE DAUGE ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Polygonal Domains ◽  
Time Harmonic ◽  
Standard Tool ◽  
Optimal Convergence Rates

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.

scholarly journals A Predictor-Corrector Finite Element Method for Time-Harmonic Maxwell’s Equations in Polygonal Domains

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Boniface Nkemzi ◽  
Jake Léonard Nkeck
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
A Priori ◽  
Electric Boundary Condition ◽  
Polygonal Domains ◽  
Time Harmonic ◽  
Predictor Corrector ◽  
Mesh Refinements

The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.

Adaptive hp-Finite Element Computations for Time-Harmonic Maxwell’s Equations

2013 ◽  
Vol 13 (2) ◽  
pp. 559-582 ◽  
Author(s):  
Xue Jiang ◽  
Linbo Zhang ◽  
Weiying Zheng
Keyword(s):  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Adaptive Methods ◽  
Adaptive Method ◽  
Adaptive Finite Element ◽  
Adaptive Finite Element Methods ◽  
Time Harmonic

AbstractIn this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell’s equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell’s equations. The hp-adaptive methods show much better performance than the h-adaptive method.

Adaptive hp-Finite Element Method for Electromagnetic Field Logging Problems

2012 ◽  
Vol 442 ◽  
pp. 109-113
Author(s):  
Zhong Hua Ma ◽  
De Jun Liu ◽  
Qi Feng
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Electromagnetic Field ◽  
Maxwell’S Equations ◽  
Degrees Of Freedom ◽  
Maxwell's Equations ◽  
Exponential Convergence ◽  
Formation Model ◽  
Weak Formulation ◽  
Element Method

A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is proposed for electromagnetic field problems. Presented in this paper are the vector expression of Maxwell's equations, three kinds of boundary conditions, stability weak formulation of Maxwell's equations, and automatic hp-adaptivity strategy. This method can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The methodology is freely available online in the form of a general public licensed C++ library Hermes (http://hpfem.org/hermes).

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