Topic 4

Andrea Bonito. Convergence of adaptive discontinuous Garlerkin methods.

ABSTRACT:
To prove the consistency, the coercivity, and the first upper bound. Another upper bound, a quasi Galerkin orthogonality, and the convergence.

Topic 3

Chen-Song Zhang. A posteriori error estimation for integro-differential operators.

ABSTRACT:
A posteriori error estimation for integro-differential operators including the upper and lower bound

Topic 2

Ricardo H. Nochetto. AFEM for the Laplace-Beltrami Operator on Surfaces.

ABSTRACT: We first introduce an AFEM for the Laplace-Beltrami operator on C1 graphs in R^d (d>=2). We derive a posteriori error estimates that account for boththe energy error in H1 and the geometric error in W^{1,1} due to approximation of the surface by a polyhedral one. We devise a marking strategy to reduce the energyand geometric errors as well as the geometric oscillation. We prove that AFEM is a contraction on a suitably scaled sum of these three quantities as soon as the geometric oscillation has been reduced beyond a threshold. The resulting AFEM converges without knowing such threshold or any constants, and starting from anycoarse initial triangulation. Several numerical experiments illustrate the theory. Finally, we introduce and analyze an AFEM for the Laplace-Beltrami operatoron parametric surfaces, thereby extending the results for graphs. Note that, due to the nature of parametric surfaces, the geometric oscillation is now measured in terms of the differences of tangential gradients rather than differences of normals as for graphs. Numerical experiments with closed surfaces are provided to illustrate the theory.

Topic 1

Long Chen. Convergence and Optimality of Adaptive Mixed Finite Element Methods.

ABSTRACT: The convergence and optimality of adaptive mixed finite element methods for second order elliptic partial differential equations is established in this paper. The main difficulty for the mixed finite element method is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error containing divergence can be bounded by data oscillation.