Abstract
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).
Highlights
In recent years one witnesses a rapidly growing interest in simultaneous spacetime methods for solving parabolic evolution equations originally introduced in [BJ89, BJ90], see e.g. [GK11, And13, UP14, Ste15, GN16, LMN16, SS17, DS18, NS19, RS18, VR18, SZ18, FK19]
With the same trial space for the primal variable, we show stability of the Galerkin discretization of this mixed system whilst utilizing a smaller trial space for the secondary variable
We provide an a priori error bound for the Galerkin discretization of the newly introduced system, and improved a priori error bounds for the methods from [And13] and [Ste15]
Summary
In recent years one witnesses a rapidly growing interest in simultaneous spacetime methods for solving parabolic evolution equations originally introduced in [BJ89, BJ90], see e.g. [GK11, And, UP14, Ste, GN16, LMN16, SS17, DS18, NS19, RS18, VR18, SZ18, FK19]. Apart from the first order system least squares formulation recently introduced in [FK19], the known well-posed simultaneous space-time variational formulations of parabolic equations in terms of partial differential operators only, so not involving non-local operators, are not coercive. As a consequence, it is non-trivial to find families of pairs of discrete trial- and test-spaces for which the resulting Petrov-Galerkin discretizations are uniformly stable. They have an equivalent interpretation as Galerkin discretizations of an extended self-adjoint mixed system, with the Riesz lift of the residual of the primal variable being the secondary variable This is the point of view we will take.
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