scholarly journals A posteriori error estimations of a coupled mixed and standard Galerkin method for second order operators

2008 ◽  
Vol 213 (1) ◽  
pp. 35-55 ◽  
Author(s):  
Emmanuel Creusé ◽  
Serge Nicaise
Keyword(s):  
Galerkin Method ◽  
Posteriori Error ◽  
Second Order ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Error Estimations ◽  
Standard Galerkin Method

On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients

2019 ◽  
Vol 40 (2) ◽  
pp. 1577-1600
Author(s):  
Gang Chen ◽  
Jintao Cui
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Elliptic Problem ◽  
Posteriori Error ◽  
Discontinuous Coefficients ◽  
Second Order ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Hybridizable Discontinuous Galerkin ◽  
Order Elliptic Problem

Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

2013 ◽  
Vol 11 (4) ◽  
Author(s):  
Josef Dalík ◽  
Václav Valenta
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Averaging Method ◽  
Order Approximation ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Second Order ◽  
Complete Analysis ◽  
A Posteriori ◽  
A Posteriori Error

AbstractAn averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x 1, x 2) at the vertices of a regular triangulation T h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Πh[u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.

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