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.

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 Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

2011 ◽  
Vol 3 (4) ◽  
pp. 493-518 ◽  
Author(s):  
Huajie Chen ◽  
Xingao Gong ◽  
Lianhua He ◽  
Aihui Zhou
Keyword(s):  
Finite Element ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Energy Functional ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Finite Element Approximations ◽  
Nonconvex Energy ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

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.

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

2016 ◽  
Vol 19 (2) ◽  
pp. 442-472
Author(s):  
Ye Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Singularly Perturbed ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

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