Space-Time Finite Volume Differencing Framework for Effective Higher-Order Accurate Discretizations of Parabolic Equations

2012 ◽  
Vol 34 (3) ◽  
pp. A1406-A1431
Author(s):  
Yaw Kyei
Keyword(s):  
Finite Volume ◽  
Parabolic Equations ◽  
Higher Order ◽  
Space Time

scholarly journals Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations

2021 ◽  
Vol 9 (8) ◽  
pp. 366-392
Author(s):  
Yaw Kyei
Keyword(s):  
Finite Volume ◽  
Parabolic Equations ◽  
Source Term ◽  
Space Time ◽  
Local Source ◽  
Source Terms ◽  
Equation Error ◽  
Diffusive Fluxes ◽  
The Right ◽  
Volume Method

A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desiredframework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through aminimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.In utilizing a local source term distribution to account for diffusive fluxes, the right minimizing quadrature weights and collocationpoints including subgrid points for the source terms may be determined and optimized for higher accuracies as well as robust higher-ordercomputational convergence. The resulting local residuals form a more complete description of the truncation errors which are then utilizedto assess the computational performances of the resulting schemes. The effectiveness of the discretization method is demonstrated by theresults and analysis of the schemes.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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
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