Seminar in Numerical Analysis: Gregor Gantner
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 , 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 , 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 , we proved linear convergence of the employed weighted-residual
error estimator with optimal algebraic rate.
 G. Gantner, D. Haberlik, and Dirk Praetorius, Adaptive IGAFEM with optimal con-
vergence rates: Hierarchical B-splines. Math. Mod. Meth. in Appl. S., Vol. 27, 2017.
 G. Gantner, Optimal adaptivity for splines in finite and boundary element methods,
PhD thesis, TU Wien, 2017.
 Michael Feischl, Gregor Gantner, Alexander Haberl, and Dirk Praetorius. Adaptive
2D IGA boundary element methods. Eng. Anal. Bound. Elem., Vol. 62, 2016
For further information about the seminar, please visit this webpage.