scholarly journals Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations

2020 ◽  
Author(s):  
Rob Stevenson ◽  
Jan Westerdiep
Keyword(s):  
Finite Element ◽  
Variational Formulation ◽  
Evolution Equations ◽  
Time Discretization ◽  
Space Time ◽  
Parabolic Problems ◽  
Related Space ◽  
Parabolic Evolution Equations ◽  
Numer Anal ◽  
Well Posed

Abstract We analyze Galerkin discretizations of a new well-posed mixed space–time variational formulation of parabolic partial differential equations. For suitable pairs of finite element trial spaces, the resulting Galerkin operators are shown to be uniformly stable. The method is compared to two related space–time discretization methods introduced by Andreev (2013, Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal., 33, 242–260) and by Steinbach (2015, Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math., 15, 551–566).

scholarly journals Space-Time Finite Element Methods for Parabolic Problems

2015 ◽  
Vol 15 (4) ◽  
pp. 551-566 ◽  
Author(s):  
Olaf Steinbach
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Evolution Equations ◽  
A Priori ◽  
Space Time ◽  
Parabolic Problems ◽  
Parabolic Evolution Equations ◽  
Priori Error Estimates ◽  
The Stability

AbstractWe propose and analyze a space-time finite element method for the numerical solution of parabolic evolution equations. This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure. The stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces. We also provide related a priori error estimates which are confirmed by numerical experiments.

scholarly journals Low-Rank Space-Time Decoupled Isogeometric Analysis for Parabolic Problems with Varying Coefficients

2019 ◽  
Vol 19 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Angelos Mantzaflaris ◽  
Felix Scholz ◽  
Ioannis Toulopoulos
Keyword(s):  
Isogeometric Analysis ◽  
Evolution Equations ◽  
Space Time ◽  
Point Of View ◽  
Low Rank ◽  
Parabolic Problems ◽  
Computational Point ◽  
Varying Coefficients ◽  
Analysis Scheme ◽  
Parabolic Evolution Equations

AbstractIn this paper we present a space-time isogeometric analysis scheme for the discretization of parabolic evolution equations with diffusion coefficients depending on both time and space variables. The problem is considered in a space-time cylinder in {\mathbb{R}^{d+1}}, with {d=2,3}, and is discretized using higher-order and highly-smooth spline spaces. This makes the matrix formation task very challenging from a computational point of view. We overcome this problem by introducing a low-rank decoupling of the operator into space and time components. Numerical experiments demonstrate the efficiency of this approach.

A note on the efficient evaluation of a modified Hilbert transformation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Olaf Steinbach ◽  
Marco Zank
Keyword(s):  
Finite Element ◽  
Evolution Equations ◽  
Time Derivative ◽  
Space Time ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Hilbert Transformation ◽  
Element Discretization ◽  
First Order ◽  
Parabolic Evolution Equations

AbstractIn this note we consider an efficient data–sparse approximation of a modified Hilbert type transformation as it is used for the space–time finite element discretization of parabolic evolution equations in the anisotropic Sobolev space H1,1/2(Q). The resulting bilinear form of the first order time derivative is symmetric and positive definite, and similar as the integration by parts formula for the Laplace hypersingular boundary integral operator in 2D. Hence we can apply hierarchical matrices for data–sparse representations and for acceleration of the computations. Numerical results show the efficiency in the approximation of the first order time derivative. An efficient realisation of the modified Hilbert transformation is a basic ingredient when considering general space–time finite element methods for parabolic evolution equations, and for the stable coupling of finite and boundary element methods in anisotropic Sobolev trace spaces.

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