Annalisa Buffa

Ericka Brivadis, Annalisa Buffa, Barbara Wohlmuth et al.

The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf-sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects. Keywords: isogeometric analysis, mortar methods, inf-sup stability, cross point modification.

Annalisa Buffa, Timon Miehling, Marco Picasso et al.

The aim of this paper is to study the well-posedness and finite element approximation of the electrostatic shear Alfvén wave equations, a coupled system of two partial differential equations arising in plasma physics as a simplified sub-model of the drift-reduced Braginskii equations. To this end, anisotropic Sobolev spaces depending on the normalized magnetic field $\b$ are introduced, together with a Poincaré-type inequality along the integral curves of $\b$, which holds under a geometric directedness condition on the magnetic field. Using these tools, existence, uniqueness, and stability of a weak solution are established via the Faedo-Galerkin method. It is also shown that the geometric condition is satisfied in tokamak and stellarator configurations. A numerical scheme is then proposed, combining Lagrange finite elements in space with a Crank-Nicolson discretization in time. The scheme is shown to conserve a discrete energy exactly in the homogeneous case, and a priori error estimates are derived in the natural energy norm. Several numerical experiments are reported in two and three space dimensions, which confirm the theoretical results and indicate that the geometric condition on the magnetic field is necessary for the invertibility of the discrete system matrix.

Annalisa Buffa, Eduardo M. Garau

In this article we develop function-based a posteriori error estimators for the solution of linear second order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We prove a global upper bound for the energy error. The theory hinges on some weighted Poincaré type inequalities, where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [Veeser and Verfürth, 2009], where the case of standard finite elements is considered. Additionally, we present numerical experiments that show the efficiency of the error estimators independently of the degree of the splines used for discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.

Pablo Antolin, Annalisa Buffa, Mathieu Fabre

In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem using an active-set strategy and for a primal space of NURBS of degree $p$ and $p-2$ for a dual space of B-Spline. A inf-sup stability is proved to ensure a good property of the method. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two- and three-dimensional and in small and large deformation demonstrate the accuracy of the proposed method.

Annalisa Buffa, Carlotta Giannelli, Philipp Morgenstern et al.

An adaptive isogeometric method based on $d$-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop [6]. In this paper we provide a complexity estimate, i.e., an estimate on how the number of mesh elements grows with respect to the number of elements that are marked for refinement by the adaptive strategy. Our estimate is in the line of the similar ones proved in the finite element context, [3,24].

Annalisa Buffa, Eduardo M. Garau

We study the local approximation properties in hierarchical spline spaces through multiscale quasi-interpolation operators. This construction suggests the analysis of a subspace of the classical hierarchical spline space (Vuong et al., 2011) which still satisfies the essential properties of the full space. The B-spline basis of such a subspace can be constructed using parent-children relations only, making it well adapted to local refinement algorithms.

Ericka Brivadis, Annalisa Buffa, Barbara Wohlmuth et al.

Mortar methods have recently been shown to be well suited for isogeometric analysis. We review the recent mathematical analysis and then investigate the variational crime introduced by quadrature formulas for the coupling integrals. Motivated by finite element observations, we consider a quadrature rule purely based on the slave mesh as well as a method using quadrature rules based on the slave mesh and on the master mesh, resulting in a non-symmetric saddle point problem. While in the first case reduced convergence rates can be observed, in the second case the influence of the variational crime is less significant.

Annalisa Buffa, Carlotta Giannelli

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.