# Functional a posteriori error estimates and adaptivity for IgA schemes

## Dr. Svetlana Matculevich

**March 28, 2017, 3:30 p.m. S2 059**

We are concern with guaranteed error control of Isogeometric Analysis (IgA) numerical approximations of elliptic boundary value problems (BVPs). The approach is discussed within the paradigm of classical *linear Poisson Dirichlet* model problem: find $u: \overline{\Omega} \rightarrow ℝ^d$ such that

$$ - \Delta_x u = f \;\; \rm{in} \;\; \Omega, \qquad u = u_D \;\; \rm{on} \;\; \partial \Omega, \qquad (1) $$

where $\Omega \subset ℝ^d$, $d \in \{1, 2, 3\}$, denotes a bounded domain having a Lipschitz boundary $\partial \Omega$, $\Delta_x$ is the Laplace operator in space, $f \in L^{2}(\Omega)$ is a given source function, and $u_D \in H_0^1 (\Sigma)$ is a given load on the boundary.

We conduct the numerical study of the functional a posteriori error estimates integrated into the IgA framework. These so-called majorants and minorants were originally introduced in [1] and later applied to different mathematical models. This type of error estimates can exploit the higher smoothness of B-Splines (NURBS, THB-Splines) 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 implies that their gradients are in $H(\Omega, \mathtt{div})$ space. Therefore, there is no need in projecting it from $\nabla u_h \in L^{2}(\Omega, ℝ^d)$ into $H(\Omega, \mathtt{div})$.

The functional approach to the error estimation in combination with IgA approximations (generated by tensor-product splines) was investigated in [2] for (1). In the current work, we test the algorithm of the majorant reconstruction suggested [2], which allows the considerable reduction of the time-costs for the error estimates calculation and, in the same time, generates guaranteed, sharp, and fully computable bounds of errors. Moreover, we combine functional error estimates with THB-Splines (the implementation provided by *G+smo*) and demonstrate their efficiency with respect to adaptive mesh generation in IgA schemes.

This is a joint work with Prof. Ulrich Langer (RICAM) and Prof. Sergey Repin (St. Petersburg Department of V.A. Steklov Institute of Mathematics RAS).

[1] S. Repin,

A posteriori error estimation for nonlinear variational problems by duality theory, Zapiski

Nauch. Sem. V. A. Steklov Math. Institute in St.-Petersburg (POMI), 243, 201--214, 1997.

[2] S. K. Kleiss and S. K. Tomar,

Guaranteed and sharp a posteriori error estimates in isogeometric analysis,

Computers & Mathematics with Applications 70 (3), 167-190, 2015.

[3] G. Kiss, C. Giannelli, U. Zore, B. Jüttler, D. Grossmann, and J. Barner,

Adaptive CAD model (re-)construction with THB-splines,

Graph. Models, 76, 273--288, 2014.