## Seminar in Numerical Analysis: Gregor Gantner (TU Wien)

**Optimal adaptivity in finite and boundary element methods**

Since the advent of isogeometric analysis (IGA) in 2005, the finite element method (FEM) and the boundary element method (BEM) with splines have become an active field of research. The central idea of IGA is to use the same functions for the approximation of the solution of the considered partial differential equation (PDE) as for the representation of the problem geometry in computer aided design (CAD). Usually, CAD is based on tensor-product splines. To allow for adaptive refinement, several extensions of these have emerged, e.g., hierarchical splines, T-splines, and LR-splines. In view of geometry induced generic singularities and the fact that isogeometric methods employ higher-order ansatz functions, the gain of adaptive refinement (resp. loss for uniform refinement) is huge.

In this talk, we first consider an adaptive FEM with hierarchical splines of arbitrary degree for linear elliptic PDE systems of second order with Dirichlet boundary condition for arbitrary dimension d≥2. We assume that the problem geometry can be parametrized over the d-dimensional unit cube. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted-residual a posteriori error estimator. In [1], we proved linear convergence of the error estimator with optimal algebraic rate.

Next, we consider an adaptive BEM with hierarchical splines of arbitrary degree for weakly-singular integral equations of the first kind that arise from the solution of linear elliptic PDE systems of second order with constant coefficients and Dirichlet boundary condition. We assume that the boundary of the geometry is the union of surfaces that can be parametrized over the unit square. Again, we propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions, where adaptivity is driven by some weighted-residual a posteriori error estimator. In [2], we proved linear convergence of the error estimator with optimal algebraic rate. In contrast to prior works, which are restricted to the Laplace model problem, our analysis allows for arbitrary elliptic PDE operators of second order with constant coefficients.

Finally, for one-dimensional boundaries, we investigate an adaptive BEM with standard splines instead of hierarchical splines. We modify the corresponding algorithm so that it additionally uses knot multiplicity increase which results in local smoothness reduction of the ansatz space. In [3], we proved linear convergence of the employed weighted-residual error estimator with optimal algebraic rate.

###### REFERENCES

[1] G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.

[2] G. Gantner, Optimal adaptivity for splines in finite and boundary element methods, PhD thesis, TU Wien, 2017.

[3] Michael Feischl, Gregor Gantner, Alexander Haberl, and Dirk Praetorius. Adaptive 2D IGA boundary element methods. Eng. Anal. Bound. Elem., Vol. 62, 2016.

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