A priori error estimates in the maximum norm are derived for Galerkin approximations to solutions of two-point boundary valué problems. The class of Galerkin spaces considered includes almost all (quasiuniform) piecewise-polynomial spaces that are used in practice. The estimates are optimal in the sense that no better rate of approximation is possible in general in the spaces employed.

Abstract In this paper we consider some rational approximations to the fractional powers of self-adjoint positive operators, arising from Gauss–Laguerre rules. We derive practical error estimates that can be used select a priori number Laguerre points necessary achieve given accuracy. also present numerical experiments show effectiveness our approaches and reliability estimates.

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 ...

We propose new locking-free finite element methods for Biot’s consolidation model by coupling nonconforming and mixed finite elements. We show a priori error estimates of semidiscrete and fully discrete solutions. The main advantage of our method is that a uniform-in-time pressure error estimate is provided with an analytic proof. In our error analysis, we do not use Grönwall’s inequality, so t...

We prove global well-posedness for the L-critical cubic defocusing nonlinear Schrödinger equation on R with data u0 ∈ H(R) for s > 1 3 . The proof combines a priori Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given...

The subject of the paper is the analysis of error estimates of the combined nite volume-nite element method for the numerical solution of a scalar nonlinear conservation law equation with a diiusion term. Nonlinear convective terms are approximated with the aid of a monotone nite volume scheme considered over the nite volume mesh dual to a triangular grid, whereas the diiusion term is discretiz...

Abstract. In this paper, we present an alternative approach to a priori L∞-error estimates for the piecewise linear nite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the nite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the sa...

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the no...