Algorithms and error bounds for multivariate piecewise constant approximation

We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-Poincar´e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions

Convergence and optimality of the adaptive Morley element method

This paper is devoted to the convergence and optimality analysis of the adaptive Morley element method for the fourth order elliptic problem. A new technique is developed to establish a quasi-orthogonality which is crucial for the convergence analysis of the adaptive nonconforming method. By introducing a new parameter-dependent error estimator and further establishing a discrete reliability property, sharp convergence and optimality estimates are then fully proved for the fourth order elliptic problem

Convergence and Optimality of Adaptive Mixed Finite Element Methods

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods 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 that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

Adaptive Spectral Galerkin Methods with Dynamic Marking

The convergence and optimality theory of adaptive Galerkin methods is almost exclusively based on the D\"orfler marking. This entails a fixed parameter and leads to a contraction constant bounded below away from zero. For spectral Galerkin methods this is a severe limitation which affects performance. We present a dynamic marking strategy that allows for a super-linear relation between consecutive discretization errors, and show exponential convergence with linear computational complexity whenever the solution belongs to a Gevrey approximation class.Comment: 20 page

Convergence of adaptive finite element methods with error-dominated oscillation

Recently, we devised an approach to a posteriori error analysis, which clarifies the role of oscillation and where oscillation is bounded in terms of the current approximation error. Basing upon this approach, we derive plain convergence of adaptive linear finite elements approximating the Poisson problem. The result covers arbritray H^-1-data and characterizes convergent marking strategies

Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm

We consider the approximation in the reaction-diffusion norm with continuous finite elements and prove that the best error is equivalent to a sum of the local best errors on pairs of elements. The equivalence constants do not depend on the ratio of diffusion to reaction. As application, we derive local error functionals that ensure robust performance of adaptive tree approximation in the reaction-diffusion norm.Comment: 21 pages, 1 figur

Approximating gradients with continuous piecewise polynomial functions

Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in terms of the local best approximations errors on elements. Thus, requiring continuity does not downgrade local approximability and discontinuous piecewise polynomials essentially do not offer additional approximation power, even for a fixed mesh. This result implies error bounds in terms of piecewise regularity over the whole admissible smoothness range. Moreover, it allows for simple local error functionals in adaptive tree approximation of gradients.Comment: 21 pages, 1 figur