Localized pointwise error estimates for mixed finite element methods

Localized pointwise error estimates for mixed finite element methods

In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show tha...

Pointwise localized error estimates for parabolic finite element equations

Sharply localized pointwise and W∞-1 estimates for finite element methods for quasilinear problems

We establish pointwise andW−1 ∞ estimates for finite element methods for a class of second-order quasilinear elliptic problems defined on domains Ω in Rn. These estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order. Our pointwise estimates are similar to and rely on results and analysis techniques of Schatz fo...

Quadrature and Schatz’s Pointwise Estimates for Finite Element Methods

We investigate numerical integration effects on weighted pointwise estimates. We prove that local weighted pointwise estimates will hold, modulo a higher order term and a negative-order norm, as long as we use an appropriate quadrature rule. To complete the analysis in an application, we also prove optimal negative-order norm estimates for a corner problem taking into account quadrature. Finall...

Higher-Order Finite Element Methods and Pointwise Error Estimates for Elliptic Problems on Surfaces

We define higher-order analogs to the piecewise linear surface finite element method studied in [Dz88] and prove error estimates in both pointwise and L2-based norms. Using the Laplace-Beltrami problem on an implicitly defined surface Γ as a model PDE, we define Lagrange finite element methods of arbitrary degree on polynomial approximations to Γ which likewise are of arbitrary degree. Then we ...