PUMA publications for /group/simtech/valueFri Jul 20 10:54:15 CEST 2018Potential Analysis1123-1171Nonlinear Neumann-Transmission Problems for Stokes and Brinkman Equations on Euclidean Lipschitz Domains382013Nonlinear domain, Brinkman boundary problem, Layer 76M Lipschitz 42B20, Stokes 46E35, 35J25, and value vorlaeufig potential operators, 76D, Fri Jul 20 10:54:15 CEST 2018Numerical Methods for Partial Differential Equations41124--1151Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons282012boundary Sobolev--Slobodetskii refinement, spaces, Galerkin problems, spaces graded Sobolev discontinuous value weighted vorlaeufig elliptic method, mesh Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev--Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (ha) of approximation as in the smooth case. � 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012Thu May 18 11:32:12 CEST 2017{233 SPRING ST, NEW YORK, NY 10013 USA}{ADVANCES IN COMPUTATIONAL MATHEMATICS}{AUG}{4}{823-842}{Collocation with WEB-Splines}{Article}{42}{2016}problem; {Collocation; WEB-spline; value Boundary Interpolation} {We describe a collocation method with weighted extended B-splines (WEB-splines) for arbitrary bounded multidimensional domains, considering Poisson's equation as a typical model problem. By slightly modifying the B-spline classification for the WEB-basis, the centers of the supports of inner B-splines can be used as collocation points. This resolves the mismatch between the number of basis functions and interpolation conditions, already present in classical univariate schemes, in a simple fashion. Collocation with WEB-splines is particularly easy to implement when the domain boundary can be represented as zero set of a weight function; sample programs are provided on the website http://www.web-spline.de. In contrast to standard finite element methods, no mesh generation and numerical integration is required, regardless of the geometric shape of the domain. As a consequence, the system equations can be compiled very efficiently. Moreover, numerical tests confirm that increasing the B-spline degree yields highly accurate approximations already on relatively coarse grids. Compared with Ritz-Galerkin methods, the observed convergence rates are decreased by 1 or 2 when using splines of odd or even order, respectively. This drawback, however, is outweighed by a substantially smaller bandwidth of collocation matrices.}