A Space-Time Adaptive Finite Element Algorithm Based on Dual Weighted Residual Methodology for Parabolic Equations

2009 ◽  
Vol 31 (5) ◽  
pp. 3324-3355 ◽  
Author(s):  
R. Bermejo ◽  
J.Carpio
Keyword(s):  
Finite Element ◽  
Parabolic Equations ◽  
Space Time ◽  
Adaptive Finite Element ◽  
Element Algorithm

scholarly journals Further results on a space-time FOSLS formulation of parabolic PDEs

2020 ◽  
Author(s):  
Gregor Gantner ◽  
Rob Stevenson
Keyword(s):  
Finite Element Method ◽  
Least Squares ◽  
Parabolic Equations ◽  
Space Time ◽  
Adaptive Finite Element Method ◽  
Order System ◽  
Well Posedness ◽  
Adaptive Finite Element ◽  
Parabolic Pdes ◽  
First Order

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer&Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven.  In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated.  The proof of the latter easily extends to a large class of least-squares formulations.

A hp adaptive finite element algorithm for fluorescence molecular tomography based on SPN model

2013 ◽  
Author(s):  
Hongbo Guo ◽  
Yuqing Hou ◽  
Xiaowei He ◽  
Jingjing Yu ◽  
Jingxing Cheng ◽  
...  
Keyword(s):  
Finite Element ◽  
Adaptive Finite Element ◽  
Fluorescence Molecular Tomography ◽  
Element Algorithm

scholarly journals Quasi-optimality of an Adaptive Finite Element Method for an Optimal Control Problem

2011 ◽  
Vol 11 (2) ◽  
pp. 107-128 ◽  
Author(s):  
Roland Becker ◽  
Shipeng Mao
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Error Estimator ◽  
Adaptive Finite Element ◽  
Posteriori Error Estimator ◽  
A Posteriori Error ◽  
Data Approximation ◽  
Element Algorithm

Abstract We prove quasi-optimality of an adaptive finite element algorithm for a model problem of optimal control including control constraints. The quasi-optimility expresses the fact that the decrease of error with respect to the number of mesh cells is optimal up to a constant. The considered algorithm is based on an adaptive marking strategy which compares a standard residualtype a posteriori error estimator with a data approximation term in each step of the algorithm in order to adapt the marking of cells.

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