scholarly journals Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

2021 ◽  
Vol 101 ◽  
pp. 107-118
Author(s):  
Rob Stevenson ◽  
Jan Westerdiep
Keyword(s):  
Parabolic Equations ◽  
Space Time ◽  
Residual Space ◽  
Spatial Operators ◽  
Minimal Residual

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.

scholarly journals Uniform controllability properties for space/time-discretized parabolic equations

2011 ◽  
Vol 118 (4) ◽  
pp. 601-661 ◽  
Author(s):  
Franck Boyer ◽  
Florence Hubert ◽  
Jérôme Le Rousseau
Keyword(s):  
Parabolic Equations ◽  
Space Time ◽  
Uniform Controllability

scholarly journals Fujita blow-up phenomena of solutions for a Dirichlet problem of parabolic equations with space-time coefficients

2021 ◽  
Author(s):  
Jiaqi Liu ◽  
Fengjie Li ◽  
Bingchen Liu
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Boundary Problem ◽  
Parabolic Equations ◽  
Blow Up ◽  
Initial Boundary ◽  
Space Time ◽  
Space Domain ◽  
Initial Boundary Problem

This paper deals with a homogeneous Dirichlet initial-boundary problem of parabolic equations with different space-time coefficients, $$u_t =\Delta u + t^{\sigma_1} u^{\alpha} + \langle x\rangle^{n} v^{p},\quad v_t =\Delta v + \langle x\rangle^{m} u^{q} + t^{\sigma_2} v^{\beta},$$ where the eight exponents are nonnegative constants and $\langle x\rangle$ is the Japanese brackets. We obtain the Fujita exponents of solutions, which are determined by the eight exponents and the dimension of the space domain. Moreover, simultaneous or non-simultaneous blow-up of the two components of blow-up solutions is discussed with or without conditions on the initial data.

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