scholarly journals A posteriori analysis of the finite element discretization of some parabolic equations

2004 ◽  
Vol 74 (251) ◽  
pp. 1117-1139 ◽  
Author(s):  
A. Bergam ◽  
C. Bernardi ◽  
Z. Mghazli
Keyword(s):  
Finite Element ◽  
Parabolic Equations ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Posteriori Analysis ◽  
A Posteriori Analysis

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

2006 ◽  
Vol 16 (02) ◽  
pp. 233-263 ◽  
Author(s):  
Z. BELHACHMI ◽  
C. BERNARDI ◽  
S. DEPARIS ◽  
F. HECHT
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Stokes Problem ◽  
A Priori ◽  
Three Dimensional ◽  
Angular Variable ◽  
A Posteriori ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Good Agreement

We consider the Stokes problem in a three-dimensional axisymmetric domain and, by writing the Fourier expansion of its solution with respect to the angular variable, we observe that each Fourier coefficient satisfies a system of equations on the meridian domain. We propose a discretization of this problem which combines Fourier truncation and finite element methods applied to each of the two-dimensional systems. We give the detailed a priori and a posteriori analyses of the discretization and present some numerical experiments which are in good agreement with the analysis.

Improved error estimates for semidiscrete finite element solutions of parabolic Dirichlet boundary control problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2898-2939 ◽  
Author(s):  
Wei Gong ◽  
Buyang Li
Keyword(s):  
Finite Element ◽  
Boundary Control ◽  
Parabolic Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Finite Element Discretization ◽  
Dirichlet Boundary Control ◽  
Element Discretization ◽  
Boundary Control Problem ◽  
Boundary Control Problems

Abstract The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal $L^p$-regularity of parabolic equations under inhomogeneous Dirichlet boundary conditions. Based on the proved regularity of the solutions, we prove ${\mathcal O}(h^{1-1/q_0-\epsilon })$ convergence for the semidiscrete finite element solutions for some $q_0>2$, with $q_0$ depending on the maximal interior angle at the corners and edges of the domain and $\epsilon$ being a positive number that can be arbitrarily small.

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