Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 8th Dec 2022, 11:11:13pm CET

 
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Session Overview
Session
Z1: FEM analysis and IGA
Time:
Tuesday, 13/July/2021:
12:00pm - 2:00pm

Session Chair: Ana Alonso Rodriguez
Virtual location: Zoom 7


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Presentations
12:00pm - 12:20pm

Flexible high order reconstructions of fields

Ludovico Bruni Bruno1, Francesca Rapetti2, Ana Alonso Rodríguez1

1Università di Trento, Italy; 2Université Côte d'Azur, France

We are interested in the interpolation on simplicial meshes of physical fields, intended as differential k-forms, in the finite element spaces of high degree trimmed polynomials (see e.g. [3]). The degrees of freedom firstly proposed in [4], that are integrals of the field under consideration on a distribution of k-simplices in each element of the mesh, yield a flexibility in the choice of the support of the degrees of freedom, that opens the way to nonuniform distributions (see [1]). It is well-know that when interpolating 0-forms, the choice of non uniform distributions of nodes where the data are assigned improves the stability.

Following this idea, we analyse the quality of the interpolation on uniform and nonuniform distributions of k-simplices for k>0 by relying on the generalised Lebesgue constant defined in [2]. Preliminary numerical results, for k=1 and k=2 in 2D and 3D, support the nonuniform choice, in agreement with the well-known nodal case. Interestingly, the sensibility of the interpolation to errors on the data, even in the uniform case, is less important than the one characterizing the nodal case. This aspect is in accordance with the computed Lebesgue constants that show a limited growth with the increase of the approximation degree and of the order of the form.

References

[1] A. Alonso Rodríguez, L. Bruni Bruno, and F. Rapetti, Towards nonuniform distributions of unisolvent weights for Whitney finite element spaces on simplices: the edge element case, submitted on December 2020, see hal-03114568.

[2] A. Alonso Rodríguez and F. Rapetti, On a generalization of the Lebesgue's constant, J. Comput. Phys., 428 (2021), 109964.

[3] S. H. Christiansen and F. Rapetti, On high order finite element spaces of differential forms, Math. Comp., 85 (2016), 517-548.

[4] F. Rapetti and A. Bossavit, Whitney forms of higher degree, SIAM J. Numer. Anal., 47 (2009), 2369-2386.



12:20pm - 12:40pm

Discrete Hodge–star operators in Isogeometric Analysis

Bernard Kapidani, Rafael Vàzquez

Ecole Polytechnique Fédérale de Lausanne

A new kind of spline geometric method approach will be presented. Its main ingredient is the use of well established sequence properties of B-spline approximations of functional spaces living on the De Rham complex [1] to construct two exact sequences of discrete differential forms: a primal sequence $\{X^k_h\}_k$ and a dual sequence $\{\tilde{X}^k_h\}_k$ with $k$ going from 0 to $n$ in $n$ space dimensions. Exploiting the recurrence relations used in defining derivatives of B-splines in one dimension, sequence properties are exactly encoded in the discrete setting. Furthermore, the two sequences can be isomorphically mapped into one another, in the sense of classical Hodge theory, if $dim(X^k_h) = dim(\tilde{X}^{n-k}_h)$. We show that this can be achieved by imposing homogeneous boundary conditions on the spaces of the primal sequence, by choosing the functions in $X^0_h$ to be an $H^1$-conforming tensor product of B-splines of degree $p>1$, and choosing degree $p-1$ for the basis of $\tilde{X}^0_h$.

Within this setup, many familiar second order partial differential equations can be finally accommodated by explicitly constructing the appropriate Hodge--star operators (which are discrete versions of constitutive relations). Several alternatives based on both global and local (of the type presented in [2]) projection operators between spline spaces will be proposed.

The appeal of our approach with respect to other geometric methods (e.g. [3]), is twofold: firstly, it exhibits high order convergence due to the involved spline spaces. Secondly, it does not rely on the geometric realization of any (topologically) dual mesh. Several numerical examples in various space dimensions will be employed to validate the central ideas of the proposed approach and compare its features with the standard Galerkin Isogeometric Analysis (IGA) approach.

References

[1] Buffa, A., Rivas, J., Sangalli, G., Vázquez, R., 2011. Isogeometric Discrete Differential Forms

in Three Dimensions. SIAM J. Numer. Anal. 49, 818–844. https://doi.org/10.1137/100786708

[2] Lee, B.-G., Lyche, T., Mørken, K., 2000. Some Examples of Quasi-Interpolants Constructed

from Local Spline Projectors, in: In Mathematical Methods in CAGD: Oslo 2000, Vanderbilt.

University Press, pp. 243–252.

[3] Tonti, E., 2001. Finite Formulation of the Electromagnetic Field. Progress In Electromagnetics

Research 32, 1–44. https://doi.org/10.2528/PIER00080101



12:40pm - 1:00pm

Dual-Primal Isogeometric Tearing and Interconnecting Methods for the Stokes problem

Jarle Sogn, Stefan Takacs

RICAM - Johann Radon Institute for Computational and Applied Mathematics, Austria

We consider the Stokes problem discretized in the Isogeometric framework for multipatch domains.

To solve the resulting system we use Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) methods. IETI-DP methods have in resent years been studied a lot for the Poisson problem. In this talk we aim to extend some the of recent results from the Poisson problem to the Stokes problem. In particular, we look at condition number bounds which are explicit in grid size h and spline degree p.

Several challenges arise since the problem now has a saddle point structure. Moreover, the standard condition that requires the average of the pressure to vanish complicates things further.

Since the pressure can be discontinuous, we only have to impose continuity for the velocity. Concerning the choice of the primal degrees of freedom, several approaches can be considered, such as, using only isolated velocity degrees of freedom as primal variables (like the corners) or using a combination of velocity and pressure.

Numerical experiments that have been carried out for the proposed IETI-DP methods will be presented.



1:00pm - 1:20pm

IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions

Rainer Schneckenleitner1, Stefan Takacs2

1Johannes Kepler University Linz, Austria; 2Austrian Academy of Sciences, Austria

We study Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the patches using a symmetric interior discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. Now, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electric motor. We show a condition number bound that coincides with the bound for the conforming case and illustrate our findings with numerical results.



1:20pm - 1:40pm

An inexact Dual-Primal Isogeometric Tearing and Interconnecting Method for continuous Galerkin discretizations

Rainer Schneckenleitner1, Stefan Takacs2

1Johannes Kepler University Linz, Austria; 2Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Austria

Recently, the authors have proven for the first time a condition number estimate for a dual-primal isogeometric tearing and interconnecting (IETI-DP) method for the Poisson problem that is explicit not only in the grid size and the subdomain diameter, but also in the spline degree. In the analysis of the IETI-DP method and the numerical experiments, the authors only have considered exact solvers for the local subproblems.

In this talk, construct and analyze a IETI-DP solver that allows the incorporation of inexact local solvers. We use the fast diagonalization (FD) method as inexact solvers for the arising local problems. We show that the condition number of the proposed IETI-DP method does not deteriorate in this case. The numerical experiments show that the new solver with FD as inexact local solvers is significantly faster than the IETI-DP methods with sparse direct local solvers. In addition, the use of FD requires less memory compared to the use of sparse direct local solvers.



1:40pm - 2:00pm

Super-Convergent Galerkin Approximations of Eigenvalues by the Adjoint-based Method

Shiqiang Xia, Bernardo Cockburn

University of Minnesota, United States of America

In this talk, we present a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations. We consider the eigenvalue as a non-linear functional of its corresponding eigenfunction and show how to extend the adjoint-based approach proposed in [1] to compute it. We illustrate the method on a second-order elliptic eigenvalue problem. Our extensive numerical results show that the approximate eigenvalues computed by our method converge with a rate of 4k + 2 when tensor-product polynomials of degree k are used for the Galerkin approximations. In contrast, eigenvalues obtained by standard finite element methods such as the mixed method or the discontinuous Galerkin method converge with a rate at most of 2k + 2. Numerical results for the classic L-shape domain and for the quantum harmonic oscillator are also presented to display the performance of the method. We also include the uncovering of a new adjoint-corrected approximation of the eigenvalues provided by the hybridizable discontinuous Galkiern method which converges with order 2k + 2, as well as preliminary results showing the possibilities of using the adjoint-correction term as an asymptotically exact a posteriori error estimate.

[1] B. Cockburn and Z. Wang, Adjoint-based, superconvergent Galerkin approximations of linear functionals, J. Sci. Comput., 73 (2017), pp. 644–666.



 
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