tangent-point radius

$r(u(y), u(x))$ which is the radius of the circle that is tangent to the curve $u$ at the point $u(y)$ and that intersects with $u$ in $u(x)$. We define evolutions via the gradient flow for $E$ within a class of arclength parametrized curves, i.e., given an initial curve $u^0 \in H^2(I;\mathbb{R}^3)$ we look for a family $u:[0,T]\to H^2(I;\mathbb{R}^3)$ such that, with an appropriate inner product $(\cdot,\cdot)_X$ on $H^2(I;\mathbb{R}^3)$,
\[
(\partial_t u, v)_X = - \, \delta E(u)[v], \quad u(0) = u^0,
\]
subject to the linearized arclength constraints
\[
[\partial_t u]' \cdot u' = 0, \quad v' \cdot u' = 0.
\]
Our numerical approximation scheme for the evolution problem is specified via a semi-implicit discretization, i.e., for a step-size $\tau>0$ and the associated backward difference quotient operator $d_t$, we compute iterates $(u^k)_{k=0,1,\dots}\subset H^2(I;\mathbb{R}^3)$ via the recursion
\[
(d_t u^k,v)_X + \kappa ([u^k]'',v'') = - \varrho \delta {\rm TP}(u^{k-1})[v]
\]
with the constraints
\[
[d_t u^k]' \cdot [u^{k-1}]' = 0, \quad v \cdot [u^{k-1}]' = 0.
\]
The scheme leads to sparse systems of linear equations in the time steps for cubic $C^1$ splines and a nodal treatment of the constraints. The explicit treatment of the nonlocal tangent-point functional avoids working with fully populated matrices and furthermore allows for a straightforward parallelization of its computation. Based on estimates for the second derivative of the tangent-point functional and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization.
We present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots. This is joint work with Philipp Reiter (University of Georgia).In the present talk we introduce a family of VEMs in the framework of incompressible fluid dynamics, more specifically the Stokes and Navier-Stokes equations. This method, that can handle general polytopal meshes, has three equivalent formulations, all yielding divergence-free velocities (which is well known to be an advantage, when compared to more traditional inf-sup stable schemes). The first one is a mixed velocity-pressure formulation. The second choice is more computationally efficient and is an immediate derivation of the first one by automatic elimination (not static condensation) of certain internal variables; although having a piecewise constant pressure, it still yields a highly accurate velocity. Finally, the third formulation is based on a suitable VEM Stokes complex (by introducing an associated highly regular virtual space) in order to obtain a smaller and definite positive system, at the price of a higher condition number.

This is a joint work with Daniele Prada and Ilaria Perugia, realized in the framework of the ERC Project CHANGE (grant agreement No 694515).

This is a joint work with Francesca Fierro and Andreas Veeser from University of Milan.

References:

[1] D. Boffi, D. Gallistl, F. Gardini, and L. Gastaldi. Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form. Mathematics of Computation, 86(307) (2017) 2213-2237

[2] D. Boffi, L. Gastaldi, R. Rodríguez, and I. Šebestová. Residual-based a posteriori error estimation for the Maxwell's eigenvalue problem. To appear in IMA Journal of Numerical Analysis. arXiv:1602.00675

[3] D. Boffi, L. Gastaldi, R. Rodríguez, and I. Šebestová. A posteriori error estimates for Maxwell's eigenvalue problem. Submitted.

[4] D. Boffi and L. Gastaldi. Adaptive finite element method for the Maxwell eigenvalue problem. arXiv:1804.02377

This is joint work with Guosheng Fu (Brown University), Francisco-Javier Sayas (University of Delaware) and Ke Shi (Old Dominion University).

shape-gradients

and expressions for those can be derived by means of (i) domain
transformations or (ii) Hadamard representation formulas.
The two approaches yield two different but equivalent formulas. Both rely on solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

(joint work with A. Paganini, S. Sargheini, and J.-Z. Li)

References:

R. Hiptmair and J.-Z. Li

, Shape derivatives in differential forms
II: Application to scattering problems

, Report 2017-24, SAM, ETH
Zürich, 2017.
R. Hiptmair and A. Paganini

, Shape optimization by pursuing diffeomorphisms

, Comput. Methods Appl. Math., 15 (2015), pp. 291–305.
R. Hiptmair, A. Paganini, and S. Sargheini

, Comparison of approximate shape gradients

, BIT Numerical Mathematics, 55 (2014),
pp. 459–485.
A. Paganini

, Numerical shape optimization with finite elements

, ETH Dissertation 23212, ETH Zürich, 2016.
S. Sargheini

, Shape Sensitivity Analysis of Electromagnetic Scattering Problems

, ETH Dissertation 23067, ETH Zürich, 2016.This research work is a joint work with Svetlana Matculevich and Sergey Repin, and was supported by the Austrian Science Fund (FWF) through the project S117-03 within the National Research Network “Geometry + Simulation”.

The framework that we provide makes it simple to create hierarchical basis with control on the overlapping. Linear independence is always desired for the well posedness of the linear systems, and to avoid redundancy. The control on the overlapping of basis functions from different levels is necessary to close theoretical arguments in the proofs of optimality of adaptive methods.

In order to guarantee linear independence, and to control the overlapping of the basis functions, some basis functions additional to those initially marked must be refined. However, with our framework and refinement procedures, the complexity of the resulting bases is under control.

More precisely, if we construct hierarchical bases $\{{\cal H}_k\}_k$ through subsequent calls to ${\cal H}_{k+1} = $Refine$({\cal H}_k,{\cal M}_k)$, where ${\cal M}_k \subset {\cal H}_k$ denotes the set of marked functions, we obtain \[ \# {\cal H}_{R} - \# {\cal H}_{0}\le C \sum_{k=0}^{R-1} \#{\cal M}_{k} , \] with a constant $C$ independent of $R$.

axioms of adaptivity

from [Carstensen et al., Comput. Math. Appl. 67, 2014] analyze under which assumptions on the a posteriori

error estimator and the mesh-refinement strategy, a mesh-refining adaptive algorithm yields convergence with optimal algebraic rates. In our talk, which is based on the recent work [Gantner et al., M3AS 27, 2017], we now address the question which properties of the FEM are sufficient to ensure that the usual weighted-residual error estimator is well-defined and satisfies the axioms of adaptivity. In particular, our analysis covers conforming FEM in the framework of isogeometric analysis with hierarchical splines.
The talk is based on joint work with Gregor Gantner (TU Wien).

In this talk, we discuss $hp$-finite elements for the discretization of primal and (dual-)mixed formulations of variational equations as well as variational inequalities resulting from contact problems. The mixed methods are based on the introduction of a flux field in the $H({\rm div})$-space. Whereas the primal approaches require some continuity of the ansatz function along the edges of the finite element mesh, the dual-mixed approaches necessitate continuity in the normal direction of the edges, which results from the discretization of $H({\rm div})$. In both cases, hybridization techniques can be applied in order to enforce the desired continuity (at least in some Gauss points) and to enable the use of Lagrange-type basis functions along with their advantageous nodal properties. A focus of the talk is on the derivation of a posteriori error estimates, where we use some post-processing reconstructions of the potential in $H^1$. Two approaches of error control are considered: In the first approach, the post-processing reconstruction is explicitly computed, whereas in the second approach, a reconstruction is applied which does not require an explicit computation. The latter enables the direct use of the discrete potential instead of its reconstruction, which significantly improves the error estimation. The applicability of the estimates within adaptive schemes is demonstrated in several numerical experiments, in which efficiency indices and convergence rates are studied.

In this talk, we will show how the underlying inf-sup theory of continuous and discretized parabolic problems provides an effective approach to the construction and rigorous analysis of parallel-in-time solvers. In particular, we consider the implicit Euler discretization of a general linear parabolic evolution equation with time-dependent self-adjoint spatial operators. We first show that the discrete system admits a similar inf-sup condition as for the underlying continuous operator. We use this to show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable in the same norms. The essential idea is that the mapping from trial functions to their optimal test functions in the inf-sup condition defines a left-preconditioner that symmetrizes the system in a stable way.

We then propose and analyse an inexact Uzawa method for the saddle-point reformulation based on an efficient parallel-in-time preconditioner. The preconditioners is non-intrusive and easy to implement in practice, since it simply combines existing spatial preconditioners with parallel Fast Fourier Transforms (FFT) in time. We prove robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes. The theoretical parallel complexity of the method then grows only logarithmically with respect to the number of time-steps, owing to the parallel FFT. Numerical experiments of large-scale parallel computations, with up to 131 072 processors and more than 2 billion unknowns, show the effectiveness of the method, along with its good weak and strong scaling properties.

Joint work with Martin Neumüller, Linz.

The talk is based on joint work with M. Zank and H. Yang.

The cost of the preconditioner is the sum of the cost of the discretized opposite order operator plus a cost that scales linearly in the number of unknowns. Thinking of the canonical example of the single layer operator, an obvious choice for the operator of opposite order is the hypersingular operator. Aiming at a preconditioner of optimal complexity, however, we wish to apply a multi-level operator instead, as the one proposed in [2].

Other than with operators of positive order, so far on locally refined meshes optimal (multi-level) preconditioners of linear complexity seem not to be available. We modify the method from [2] such that it applies on locally refined meshes.

References:

[1] A. Buffa and S. H. Christiansen. A dual finite element complex on the barycentric refinement.

Math. Comp.

, 76(260):1743–1769, 2007.
[2] J. H. Bramble, J. E. Pasciak, and P. S. Vassilevski. Computational scales of Sobolev norms with application to preconditioning.

Math. Comp.

, 69(230):463–480, 2000.
[3] R. Hiptmair. Operator preconditioning.

Comput. Math. Appl.

, 52(5):699–706, 2006.
[4] O. Steinbach and W. L. Wendland. The construction of some efficient preconditioners in the boundary element method.

Adv. Comput. Math.

, 9(1-2):191–216, 1998.
Numerical treatment of boundary integral equations.
[5] R. P. Stevenson and R. van Venetië. Optimal preconditioning for problems of negative order, 2018, arXiv:1803.05226. Submitted.

We consider a particular class of $C^0$-smooth planar multi-patch B-spline parametrizations, satisfying specific geometric continuity constraints, which allow the construction of $C^1$-smooth isogeometric spaces with optimal approximation properties. We characterize those spaces and study their properties and suitability for isogeometric analysis. Moreover, we discuss the construction of a basis for the $C^1$-smooth spaces, which generalizes the idea of the Argyris finite element to tensor-product spline patches.

The work presented here is based on a collaboration with Mario Kapl and Giancarlo Sangalli.

In the context of finite elements, a dual finite element complex, defined in a new mesh constructed by barycentric refinement, was introduced by Buffa and Christiansen in [2]. In the present paper we develop a dual spline complex for isogeometric methods. Compared to the dual complex in [2], the dual spline complex we present is much easier to construct, thanks to the tensor-product structure of B-splines. We will also show preliminary work on the generalization of the construction to domains formed by the union of several patches, that would generalize our setting to arbitrary topology.

References:

[1] A. Buffa, G. Sangalli, R. Vázquez, Isogeometric analysis in electromagnetics: B-splines approximation, Comput. Methods Appl. Mech. Engrg. 199 (2010), 1143-1152.

[2] A. Buffa and S. Christiansen, A dual finite element complex on the barycentric refinement, Math. Comp. 76 (2007) 1743-1769.