afem

Convergence and optimality of AFEM

Adaptive methods are now widely used in scientific and engineering computation to optimize the relation between accuracy and computational labor (degrees of freedom). Understanding the convergence of adaptive finite element methods (AFEM) and the approximation produced by AFEM has been an active topic in recent years. The following list my contribution to this field.

• L. Chen, M.J. Holst, and J. Xu. The Finite Element Approximation of the Nonlinear Poisson-Boltzmann Equation. SIAM Journal on Numerical Analysis, 45(6): 2298-2320, 2007.

This work apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson-Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for this equation.

• L. Chen, M.J. Holst, and J. Xu. Convergence and optimality of adaptive mixed finite element methods. 78: 35--53, Mathematics of Computation, 2009.

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. A discrete stability sresult is established and used to get a quasi-orthogonality and localized discrete upper bound.

• L. Zhong, S. Shu, L. Chen, and J. Xu. Convergence of adaptive edge finite element methods for H(curl)-elliptic problems. Numerical Linear Algebra with Applications. 17(2-3):415-432,2010.

The standard Adaptive Edge Finite Element Method (AEFEM), using first/second family Nedelec edge elements with any order, for the three dimensional H(curl)-elliptic problems with variable coefficients is shown to be convergent for the sum of the energy error and the scaled error estimator. The special treatment of the data oscillation and the interior node property are removed from the proof. Numerical experiments indicate that the adaptive meshes and the associated numerical complexity are quasi-optimal.

• L. Chen. Optimal Rate of Convergence in $L^2$-Norm of An Adaptive Finite Element Method for Elliptic Equations. Submitted,2009.

An adaptive finite element algorithm to control the error in $L^2$-norm is developed for second order elliptic equations. It complements the standard adaptive finite element method with a procedure to control the mesh size according to the a priori information on the second derivative of the solution.

• L. Zhong, L. Chen, S. Shu, G. Wittum, and J. Xu. Quasi-optimal convergence of adaptive edge finite element methods for three dimensional indefinite time-harmonic Maxwell's equations. Submitted, 2009.

We extend the theory of AFEM for Poisson-type equations to indefinite Maxweel's equations. To deal with non-symmetric parts, we give an L2 estimate which is quite different from and much more difficult than that of elliptic equations since the standard duality approach does not work. We prove a quasi-optimal asymptotic rate of convergence for adaptive procedure. The crucial technique is a localized upper bound.