Goal-oriented error estimation for transient parabolic problems
The paper focuses on the estimation of errors present when solving transient parabolic problems with finite element methods (FEMs). The authors consider the parabolic equation \(\dot u-\nabla \cdot (\alpha \nabla u)=f\) in \( \Omega \times I\) where \(\Omega \) is a bounded domain of \(\mathbb{R}^{d}\), \( d=1,2,3\), and \(I=(0,T)\). An initial data \(u(x,0)=u_{0}(x)\) is given and homogeneous Dirichlet boundary conditions are imposed on \(\partial \Omega \times I\). When applying a FEM discretization, a weak formulation of the problem is written as: find \(u^{H}\in \mathcal{V}^{H}\) such that \((\dot{u}^{H}(t),v)+a(u^{H}(t),v)=l(v)\) for every \(v\in \mathcal{V}^{H}\), where \( \mathcal{V}^{H}\) is a subspace of \(\mathcal{H}_{0}^{1}(\Omega )\). A time discretization is introduced through a time grid decomposing \(I\) in subintervals \(I_{n}\) and considering a space \(\mathcal{V}_{\Delta t}=\{v:I\rightarrow \mathbb{R};v\mid _{I_{n}}\in \mathcal{P}_{q}(I_{n})\}\), where \(\mathcal{P}_{q}(I_{n})\) is the space of polynomials of degree less than \(q\) in \(I_{n}\). A discontinuous Galerkin procedure is then used which combines \(\mathcal{V}^{H}\) and \(\mathcal{V}_{\Delta t}\) in \(\mathcal{V} _{\Delta t}^{H}\) and consists to write a weak formulation of the problem in the space \(\mathcal{V}_{\Delta t}^{H}\): find \(u^{H}\in \mathcal{V}_{\Delta t}^{H}\) such that \(B(u^{H},v)=L(v)\) for every \(v\in \mathcal{V}_{\Delta t}^{H}\). This weak formulation involves the jump of \(u^{H}\) at \(t^{n}\). In order to compute the errors of some quantities called of interest, the authors write the weak formulation of the original problem: find \(u\in \mathcal{V}\) such that \(B(u,v)=L(v)\) for every \(v\in \mathcal{V}\). The authors consider quantities of the kind \(J(u)=(j(u(.,T)))\). They introduce a dual problem \(B(v,\varphi )=J(v)\) for every \(v\in \mathcal{V}\) and derive the problem whose \(\varphi \) is a solution. The idea of the paper consists first to insert the error \(u-u^{H}\) in the last variational formulation and to obtain a representation of this error (weak residual in the primal problem). Then the authors explain how an error estimation allows to refine the space or time grids. The last part of the paper presents examples where these tools can be used.
scientific article; zbMATH DE number 5552314