Higher order local accuracy by averaging in the finite element method
James H. Bramble,A. H. Schatz +1 more
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In this paper, the authors describe the class of finite element subspaces and explain the main result on the accuracy of K h * u h, where K h is a fixed function, u h represents local averages, and * denotes convolution.Abstract:
This chapter describes the class of finite element subspaces and explains the main result on the accuracy of K h * u h ,where K h is a fixed function, u h represents local averages, and * denotes convolution. The function K h has the following properties: (1) K h has small support; (2) K h is independent of the specific choice of S h or the operator L; (3) K h * u h is easily computable from u h ; and (4) K h * u h approximates u to higher order than does u h . The chapter also discusses on notation, subspaces and the construction of K h .read more
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The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique
O. C. Zienkiewicz,J. Z. Zhu +1 more
TL;DR: In this article, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes, which has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems.
Review Article Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: The theoretical and algorithmic aspects of the Runge–Kutta discontinuous Galerkin methods are reviewed and several applications including nonlinear conservation laws, the compressible and incompressible Navier–Stokes equations, and Hamilton–Jacobi-like equations are shown.
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Runge-Kutta discontinuous Galerkin methods for convection-dominated problems
Bernardo Cockburn,Chi-Wang Shu +1 more
TL;DR: The Runge-Kutta discontinuous Galerkin (RKDG) method as discussed by the authors is one of the state-of-the-art methods for non-linear convection-dominated problems.
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An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems
Paul Castillo,Bernardo Cockburn +1 more
TL;DR: This paper presents the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem and shows that, for stabilization parameters of order one, the L2-norm of the gradient and the L1- norm of the potential are of order k and k+1/2, respectively.
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Non-homogeneous boundary value problems and applications
TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
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The finite element method with Lagrangian multipliers
TL;DR: In this article, the Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions.