For a boundary integral formulation of the 2D Laplace equation with mixed conditions, we consider an adaptive Galerkin BEM based on [Formula: see text]-type error estimator. We include resolution Dirichlet, Neumann, and volume data into algorithm. In particular, implementation developed algorithm has only to deal discrete operators. prove that proposed scheme leads sequence solutions, for which corresponding estimators tend zero. Under saturation assumption non-perturbed problem is observed...

For the simple layer potential $V$ associated with three-dimensional (3D) Laplacian, we consider weakly singular integral equation $V\phi=f$. This is discretized by lowest-order Galerkin boundary element method. We prove convergence of an $h$-adaptive algorithm that driven a weighted residual error estimator. Moreover, identify approximation class for which adaptive converges quasi-optimally respect to number elements. In particular, mesh refinement superior uniform refinement.

Abstract We study a first-order system formulation of the (acoustic) wave equation and prove that operator this is an isomorphism from appropriately defined graph space to $L^{2}$. The results rely on well-posedness stability weak ultraweak second-order equation. As application, we define analyze space-time least-squares finite element method for solving Some numerical examples one- two-dimensional spatial domains are presented.

Abstract. We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for weakly-singular integral equation in 2D. The mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator not only reliable, but under some regularity assumptions on given data also efficient locally refined meshes, we characterize approximation class terms Galerkin only. In particular, yields no strategy can do better, thus an optimal choice to...

For first-order discretizations of the integral fractional Laplacian, we provide sharp local error estimates on proper subdomains in both $H^1$-norm and localized energy norm. Our have form a best approximation plus global measured weaker

We report on the Matlab program package HILBERT. It provides an easily-accessible implementation of lowest order adaptive Galerkin boundary element methods for numerical solution Poisson equation in 2D. The library was designed to serve several purposes: stable integral operators may be used research code. framework ensures usability lectures or scientific computing. Finally, we emphasize use adaptivity as general concept and particular. In this work, summarize recent analytical results...

We present and analyze a discontinuous Petrov--Galerkin method with optimal test functions for reaction-dominated diffusion problem in two three space dimensions. start an ultraweak formulation that comprises parameters $\alpha$, $\beta$ to allow general $\varepsilon$-dependent weightings of field variables ($\varepsilon$ being the small parameter). Specific values $\alpha$ imply robustness method, is, quasi-optimal error estimate constant is independent $\varepsilon$. Moreover, these lead...

Abstract We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. complement first-order optimality condition recently developed space-time variational formulation equations which is coercive in energy norm, and Lagrange multiplier. Our final fulfills Babuška–Brezzi conditions on continuous as well discrete level, without restrictions. Consequently, we can allow final-time desired states, obtain an...

We consider fractional Sobolev spaces Hθ, θ ∈ (0,1), on 2D domains and H1-conforming discretizations by globally continuous piecewise polynomials a mesh consisting of shaperegular triangles quadrilaterals. prove that the norm obtained from interpolating between discrete space equipped with L2-norm one hand H1-norm other is equivalent to corresponding interpolation norm, norm-equivalence constants are independent meshsize polynomial degree. This characterization then used show an inverse...

We prove local inverse-type estimates for the four non-local boundary integral operators associated with Laplace operator on a bounded d-dimensional Lipschitz domain Omega d >= 2 piecewise smooth boundary. For polynomial ansatz spaces and = or 3, inverse are explicit in both mesh width approximation order. An application to efficiency posteriori error estimation element methods is given.

Minimum residual methods such as the least-squares finite element method (FEM) or discontinuous Petrov--Galerkin (DPG) with optimal test functions usually exclude singular data, e.g., non-square-integrable loads. We consider a DPG and FEM for Poisson problem. For both we analyze regularization approaches that allow use of $H^{-1}$ loads also study case point all cases prove appropriate convergence orders. present various numerical experiments confirm our theoretical results. Our approach...

Abstract We consider the adaptive lowest‐order boundary element method based on isotropic mesh refinement for weakly‐singular integral equation three‐dimensional Laplacian. The proposed scheme resolves both, possible singularities of solution as well given data. implementation thus only deals with discrete operators, that is, matrices. prove usual mesh‐refining algorithm drives corresponding error estimator to zero. Under an appropriate saturation assumption which is observed empirically,...

In the context of adaptive finite element method (FEM), ZZ-error estimators named after Zienkiewicz and Zhu (1987) [52] are mathematically well-established widely used in practice. this work, we propose analyze ZZ-type error for boundary (BEM). We consider weakly singular hyper-singular integral equations prove, particular, convergence related mesh-refining algorithms. Throughout, theoretical findings underlined by numerical experiments.