A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations (Q5944616)

The goal of this work is to establish a posteriori error estimators for semidiscrete finite element approximations of parabolic partial differential equations. The approach taken is to make use of techniques used to obtain estimators for elliptic problems. NEWLINENEWLINENEWLINETwo model problems are considered: diffusion or heat conduction, and a viscoelasticity problem of the form NEWLINE\[NEWLINE \gamma \frac{\partial u}{\partial t} - \square u = f(t) NEWLINE\]NEWLINE together with standard boundary and initial conditions, where \(\square\) is the elasticity operator NEWLINE\[NEWLINE \square u = \lambda \nabla \text{div} u + 2\mu \text{div} (\nabla u) NEWLINE\]NEWLINE with \(\lambda\) and \(\mu\) being the Lamé constants, and \(u\) the displacement vector. NEWLINENEWLINENEWLINEIn both cases weak versions of the problems are formulated. For the heat conduction problem this takes the form NEWLINE\[NEWLINE B(u,v) = \ell (v) NEWLINE\]NEWLINE where NEWLINE\[NEWLINE B(u,v) = \gamma \int_0^T \int_0^t \left( \frac{\partial u}{\partial t}_{L^2(\Omega)} dt,d\tau + \int_0^T \int_0^t (\nabla u , \nabla v)_{L^2(\Omega)} dt,d\tau\right)NEWLINE\]NEWLINE and \(\ell\) is the linear functional associated with the source term. The weak formulation of the viscoelasticity problem is similar. The solution is sought in \(U := \{ \phi \in L^2((0,T); {\mathcal E}(\Omega)),\;\;\partial \phi / \partial t \in L^2(Q_T),\;\;\phi (x,0) = 0 \}\), for all functions \(v\) in the space \(V := L^2((0,T); {\mathcal E}(\Omega))\). Here \({\mathcal E}\) is the energy space, with norm the \(H^1\)-seminorm. NEWLINENEWLINENEWLINEStandard discrete spaces \(U^h\) and \(V^h\) are introduced, and an estimate for the error is then obtained in terms of the residual. By a lengthy and technical, but not inherently complicated analysis, computable lower and upper bounds on the error are obtained. It is shown that any good a posteriori estimator for the elliptic problem yields a correspondingly good estimator for the parabolic problem. NEWLINENEWLINENEWLINETwo examples are presented. The first is a one-dimensional heat conduction problem for which a closed-form solution exists. The effectivity index is tracked as a means of measuring the effectiveness of the various estimators. The second example is a two-dimensional problem in viscoelasticity. This takes the form of a thin cracked plate on a viscoelastic half-space. The key challenge here is that the estimators should be robust in spite of the singularity in the domain. This is found to be the case here.