A posteriori error estimates space-time in Isogeometric Analysis of parabolic problems

Dr. Svetlana Matculevich

Nov. 4, 2016, 9:15 a.m. S2 054

We are concern with guaranteed error control of space-time Isogeometric Analysis (IgA) numerical approximations of parabolic evolution equations in fixed and moving spatial computational domains. The approach is discussed within the paradigm of classical linear parabolic initial-boundary value problem (I-BVP) as model problem: find $u: \overline{Q} \rightarrow ℝ^d$ such that

\begin{alignat}{2} \partial_t u - \Delta_x u & = f \;\; {\rm in} \;\; Q, \qquad u = 0 \;\; {\rm on} \;\; \Sigma, \qquad u = u_0 \;\; {\rm on} \;\; \overline{\Sigma}_0, \label{eq:equation} \qquad (1) \end{alignat}

where $\overline{Q} := Q \cup \partial Q$, $Q := \Omega \times (0, T)$, denote the space-time cylinder with a bounded domain $\Omega \subset ℝ^d$, $d \in {1, 2, 3}$, having a Lipschitz boundary $\partial \Omega$, and $(0, T)$ is a given time interval, $0 < T < +\infty$. Here, the cylindrical surface is defined as $\partial Q := \Sigma \cup \overline{\Sigma}_{0} \cup \overline{\Sigma}_{T}$ with $\Sigma = \partial \Omega \times (0, T)$, $\Sigma_{0} = \Omega \times {0}$ and $\Sigma_{T} = \Omega \times {T}$.

Following the spirit of [1], we consider a stable IgA space-time scheme for variation formulation of (1), which is obtained by testing it with auxilary function $v_h + \delta_h , \partial_t v_h$, $\delta_h = \theta h$, $v_h \in V_{0h} \subset H^{1}_{0, \underline{0}}(Q),$ where $\theta$ is a positive constant and $h$ is the global mesh-size parameter (with mesh denoted by $\mathcal{K}_h$). The parameter $\delta_h$ in discrete bilinear form can be also localized such that $\delta_{K} = \theta , h_K$, where $h_K := {\rm diam} (K)$, $K \in \mathcal{K}_h$ is the local mesh-size parameter. It is shown that for both cases the obtained discrete bilinear forms are $V_{0h}$-coercive on the IgA space with respect to corresponding discrete energy norms, which together with boundedness property, consistency and approximation results for the IgA spaces provides an a priori discretization error estimates.

Finally, we derive the functional a posteriori error estimates for the discussed schemes (see, e.g., [2]), which apart from the quantitatively efficient indicators provides the reliable and sharp error estimates. This type of error estimates can exploit the higher smoothness of NURBS basis functions to its advantage. Since the obtained approximations are generally $C^{p-1}$-continuous (provided that the inner knots have the multiplicity $1$), this automatically provides that its gradients are in $H(\Omega, {\rm div})$ space. Therefore, there is no need in projecting it from $\nabla u_h \in L^{2}(\Omega, ℝ^d)$ into $H(\Omega, {\rm div})$.

[1] U. Langer, S. Moore, and M. Neumüller,
Space-time isogeometric analysis of parabolic evolution equations,
Comput. Methods Appl. Mech. Engrg., 2016, 306: 342–363.
[2] S. I. Repin,
Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation,
Rend. Mat. Acc. Lincei, 13(9):121–133, 2002.