An Optimal Order CG-DG Space-Time Discretization Method for Parabolic Problems

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).

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An efficient full space-time discretization method for subject-specific hemodynamic simulations of cerebral arterial blood flow with distensible wall mechanics

Journal of Biomechanics ◽

10.1016/j.jbiomech.2019.02.014 ◽

2019 ◽

Vol 87 ◽

pp. 37-47

Author(s):

Chang Sub Park ◽

Ali Alaraj ◽

Xinjian Du ◽

Fady T. Charbel ◽

Andreas A. Linninger

Keyword(s):

Blood Flow ◽

Time Discretization ◽

Space Time ◽

Arterial Blood Flow ◽

Discretization Method ◽

Arterial Blood ◽

Subject Specific ◽

Hemodynamic Simulations

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A sparse grid space-time discretization scheme for parabolic problems

Computing ◽

10.1007/s00607-007-0241-3 ◽

2007 ◽

Vol 81 (1) ◽

pp. 1-34 ◽

Cited By ~ 24

Author(s):

M. Griebel ◽

D. Oeltz

Keyword(s):

Time Discretization ◽

Space Time ◽

Sparse Grid ◽

Parabolic Problems ◽

Discretization Scheme

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A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2018059 ◽

2019 ◽

Vol 53 (2) ◽

pp. 523-549 ◽

Cited By ~ 2

Author(s):

Dimitra Antonopoulou ◽

Michael Plexousakis

Keyword(s):

Error Bounds ◽

A Priori Estimates ◽

American Mathematical Society ◽

Parabolic Type ◽

Space Time ◽

Parabolic Problems ◽

Optimal Order ◽

Cylindrical Domain ◽

A Posteriori ◽

Time Space

This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

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