Adaptive boundary element methods with convergence rates

Adaptive Finite Element Methods with convergence rates

Adaptive boundary element methods for optimal convergence of point errors

One particular strength of the boundary element method is that it allows for a high-order pointwise approximation of the solution of the related partial differential equation via the representation formula. However, the high-order convergence and hence accuracy usually suffers from singularities of the Cauchy data. We propose two adaptive mesh-refining algorithms and prove their quasi-optimal c...

Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, Part II: Hyper-singular integral equation

We analyze an adaptive boundary element method with fixed-order piecewise polynomials for the hyper-singular integral equation of the Laplace-Neumann problem in 2D and 3D which incorporates the approximation of the given Neumann data into the overall adaptive scheme. The adaptivity is driven by some residual-error estimator plus data oscillation terms. We prove convergence even with quasi-optim...

Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, Part I: Weakly-singular integral equation

We analyze an adaptive boundary element method for Symm’s integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data g into the adaptive scheme. We prove quasi-optimal convergence rates for any H-stable projection used for data approximation.

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.