scholarly journals Approximation Properties of a Gradient Recovery Operator Using a Biorthogonal System

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Bishnu P. Lamichhane ◽  
Adam McNeilly
Keyword(s):  
Finite Element ◽  
Biorthogonal System ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Efficient Computation ◽  
Oblique Projection ◽  
Approximation Properties ◽  
Gradient Recovery ◽  
Element Space ◽  
Finite Element Space

A gradient recovery operator based on projecting the discrete gradient onto the standard finite element space is considered. We use an oblique projection, where the test and trial spaces are different, and the bases of these two spaces form a biorthogonal system. Biorthogonality allows efficient computation of the recovery operator. We analyze the approximation properties of the gradient recovery operator. Numerical results are presented in the two-dimensional case.

An improved finite element space for discontinuous pressures

2010 ◽  
Vol 199 (17-20) ◽  
pp. 1019-1031 ◽  
Author(s):  
Roberto F. Ausas ◽  
Fabrício S. Sousa ◽  
Gustavo C. Buscaglia
Keyword(s):  
Finite Element ◽  
Element Space ◽  
Finite Element Space

A Unified Topological Layer for Finite Element Space Discretization

2010 ◽  
Author(s):  
Franz Stimpfl ◽  
Josef Weinbub ◽  
René Heinzl ◽  
Philipp Schwaha ◽  
Siegfried Selberherr ◽  
...  
Keyword(s):  
Finite Element ◽  
Element Space ◽  
Finite Element Space ◽  
Space Discretization

scholarly journals A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains

2020 ◽  
Author(s):  
C M Elliott ◽  
T Ranner
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Rates Of Convergence ◽  
Time Dependent ◽  
Unified Theory ◽  
Approximation Properties ◽  
Abstract Theory ◽  
Partial Differential ◽  
Element Space

Abstract We develop a unified theory for continuous-in-time finite element discretizations of partial differential equations posed in evolving domains, including the consideration of equations posed on evolving surfaces and bulk domains, as well as coupled surface bulk systems. We use an abstract variational setting with time-dependent function spaces and abstract time-dependent finite element spaces. Optimal a priori bounds are shown under usual assumptions on perturbations of bilinear forms and approximation properties of the abstract finite element spaces. The abstract theory is applied to evolving finite elements in both flat and curved spaces. Evolving bulk and surface isoparametric finite element spaces defined on evolving triangulations are defined and developed. These spaces are used to define approximations to parabolic equations in general domains for which the abstract theory is shown to apply. Numerical experiments are described, which confirm the rates of convergence.

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