Convergence rates for adaptive finite elements

Convergence rates for adaptive finite elements

In this article we prove that it is possible to construct, using newestvertex bisection, meshes that equidistribute the error in H-norm, whenever the function to approximate can be decomposed as a sum of a regular part plus a singular part with singularities around a finite number of points. This decomposition is usual in regularity results of Partial Differential Equations (PDE). As a conseque...

Adaptive Finite Element Methods with convergence rates

A Goal-Oriented Adaptive Finite Element Method with Convergence Rates

An adaptive finite element method is analyzed for approximating functionals of the solution of symmetric elliptic second order boundary value problems. We show that the method converges and derive a favorable upper bound for its convergence rate and computational complexity. We illustrate our theoretical findings with numerical results.

On Multigrid Convergence for Quadratic Finite Elements

Quadratic and higher order finite elements are interesting candidates for the numerical solution of (elliptic) partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. While the systems of equations that arise from the discretisation of the underlying PDEs are often solved by iterative schemes like preconditioned Krylow-space metho...

Adaptive Finite Elements for State-Constrained Optimal Control Problems - Convergence Analysis and A Posteriori Error Estimation