Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D

2018 ◽  
Vol 25 (3) ◽  
pp. e2143 ◽  
Author(s):  
Olaf Steinbach ◽  
Huidong Yang
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Multigrid Methods ◽  
Space Time ◽  
Algebraic Multigrid ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Algebraic Multigrid Methods

scholarly journals Fully discrete finite element data assimilation method for the heat equation

2018 ◽  
Vol 52 (5) ◽  
pp. 2065-2082 ◽  
Author(s):  
Erik Burman ◽  
Jonathan Ish-Horowicz ◽  
Lauri Oksanen
Keyword(s):  
Finite Element ◽  
Initial Data ◽  
Heat Equation ◽  
Time Derivative ◽  
Classical Problem ◽  
Classical Case ◽  
Space Time ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Discretization

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty on the H2-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen [Numer. Math. 139 (2018) 505–528], combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from t = 0, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the L2-norm at the final time.

Sign in / Sign up

Export Citation Format

Share Document