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- Alfred H. Schatz
- SIAM J. Numerical Analysis
- 2000

Existence, uniqueness and error estimates for Ritz-Galerkin methods are o discussed in the case where the associated bilinear form satisfies a Carding type inequality, i.e., it is indefinite in a certain way. An application to the finite element method is given. In this note, we would like to discuss existence, uniqueness and estimates over the whole domain… (More)

For a model convection-dominated singularly perturbed convection-diffusion problem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost hi/A… (More)

- Wolfgang Hoffmann, Alfred H. Schatz, Lars B. Wahlbin, Gabriel Wittum
- Math. Comput.
- 2001

A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on… (More)

- Alfred H. Schatz, Junping Wang
- Math. Comput.
- 1996

New uniform error estimates are established for finite element approximations uh of solutions u of second-order elliptic equations Lu = f using only the regularity assumption ‖u‖1 ≤ c‖f‖−1. Using an Aubin–Nitsche type duality argument we show for example that, for arbitrary (fixed) ε sufficiently small, there exists an h0 such that for 0 < h < h0 ‖u− uh‖0 ≤… (More)

Let ÍÍ be a polygonal domain in the plane and Sy(£l) denote the finite element space of continuous piecewise polynomials of degree < r — 1 (r > 2) defined on a quasi-uniform triangulation of ii (with triangles roughly of size h). It is shown that if un e Sy(Sl) is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the… (More)

- Alfred H. Schatz, Lars B. Wahlbin
- Math. Comput.
- 2004

We extend results from Part I about estimating gradient errors elementwise a posteriori, given there for quadratic and higher elements, to the piecewise linear case. The key to our new result is to consider certain technical estimates for differences in the error, e(x1)− e(x2), rather than for e(x) itself. We also give a posteriori estimators for second… (More)

- Mohammad Asadzadeh, Alfred H. Schatz, Wolfgang L. Wendland
- Math. Comput.
- 2009

This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in RN , N ≥ 2. The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather… (More)

- Johnny Guzmán, Dmitriy Leykekhman, Jürgen Roßmann, Alfred H. Schatz
- Numerische Mathematik
- 2009

A model second-order elliptic equation on a general convex polyhedral domain in three dimensions is considered. The aim of this paper is twofold: First sharp Hölder estimates for the corresponding Green’s function are obtained. As an applications of these estimates to finite element methods, we show the best approximation property of the error in W 1 ∞. In… (More)

- Alan Demlow, Johnny Guzmán, Alfred H. Schatz
- Math. Comput.
- 2011

Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global “pollution” term that measures the influence of solution quality from outside the domain of interest and is… (More)