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.

scholarly journals CONVERGENT ADAPTIVE FINITE ELEMENT METHODS FOR PHOTONIC CRYSTAL APPLICATIONS

2012 ◽  
Vol 22 (10) ◽  
pp. 1250028
Author(s):  
STEFANO GIANI
Keyword(s):  
Finite Element ◽  
Photonic Crystal ◽  
Eigenvalue Problems ◽  
Adaptive Methods ◽  
Discontinuous Coefficients ◽  
Adaptive Method ◽  
Adaptive Finite Element Method ◽  
Global Maximum ◽  
Adaptive Finite Element ◽  
Adaptive Finite Element Methods

We prove the convergence of an adaptive finite element method for computing the band structure of two-dimensional periodic photonic crystals with or without compact defects in both the TM and TE polarization cases. These eigenvalue problems involve non-coercive elliptic operators with discontinuous coefficients. The error analysis extends the theory of convergence of adaptive methods for elliptic eigenvalue problems to photonic crystal problems, and in particular deals with various complications which arise essentially from the lack of coercivity of the elliptic operator with discontinuous coefficients. We prove the convergence of the adaptive method in an oscillation-free way and with no extra assumptions on the initial mesh, beside the conformity and shape regularity. Also we present and prove the convergence of an adaptive method to compute efficiently an entire band in the spectrum. This method is guaranteed to converge to the correct global maximum and minimum of the band, which is a very useful piece of information in practice. Our numerical results cover both the cases of periodic structures with and without compact defects.

Error Analysis and Adaptive Methods of Least Squares Nonconforming Finite Element for the Transport Equations

2016 ◽  
Vol 8 (5) ◽  
pp. 871-886
Author(s):  
Huipo Liu ◽  
Shuanghu Wang ◽  
Hongbin Han
Keyword(s):  
Finite Element ◽  
Least Squares ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Adaptive Methods ◽  
Transport Equations ◽  
Adaptive Finite Element Method ◽  
Nonconforming Finite Element ◽  
Adaptive Finite Element ◽  
The Stability

AbstractIn this paper, we consider a least squares nonconforming finite element of low order for solving the transport equations. We give a detailed overview on the stability and the convergence properties of our considered methods in the stability norm. Moreover, we derive residual type a posteriori error estimates for the least squares nonconforming finite element methods underH–1-norm, which can be used as the error indicators to guide the mesh refinement procedure in the adaptive finite element method. The theoretical results are supported by a series of numerical experiments.

ADAPTIVE FINITE ELEMENT METHODS FOR THE OBSTACLE PROBLEM

1992 ◽  
Vol 02 (04) ◽  
pp. 483-487 ◽  
Author(s):  
CLAES JOHNSON
Keyword(s):  
Finite Element ◽  
Posteriori Error ◽  
Adaptive Method ◽  
Unilateral Constraint ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Adaptive Finite Element Methods ◽  
A Posteriori Error

We prove on a posteriori error estimate for a finite element method for an elliptic variational inequality with unilateral constraint. We formulate a corresponding adaptive method and prove reliability and efficiency of the method.

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.

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