Session I.1 - Multiresolution and Adaptivity in Numerical PDEs
Talks
Monday, June 12, 14:00 ~ 15:00
A posteriori error estimation in the maximum norm
Alan Demlow
Texas A&M University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Finite element error estimation in non-energy norms is important in many applications, but presents significant technical difficulties beyond those encountered when estimating energy norm errors. In this talk I will survey techniques for estimating errors a posteriori in the maximum norm for finite element, including recent results on estimating maximum errors for singularly perturbed convection-diffusion equations. Time permitting, I will also discuss challenges associated with proving convergence of adaptive finite element algorithms for controlling maximum errors.
Monday, June 12, 15:00 ~ 15:30
AFEM for the fractional Laplacian
Markus Melenk
TU Wien, AUstria - This email address is being protected from spambots. You need JavaScript enabled to view it.
For the discretization of the integral fractional Laplacian $(-\Delta)^s$, $0 \lt s \lt 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$-regularity of the residual in the regime $3/4 \lt s \lt 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator in the framework of [Carstensen, Feischl, Page, Praetorius, Axioms of adaptivity, CAMWA (2018)] . Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian. These local inverse estimates have further applications. For example, they underlie the proof that multilevel diagonal scaling leads to uniformly bounded condition numbers even in the presence of locally refined meshes. In the second part of the talk, we will present one such optimal multilevel diagonal scaling preconditioner.
Joint work with Bjoern Bahr (TU Wien), Markus Faustmann (TU Wien), Maryam Parvizi (Universitaet Hannover) and Dirk Praetorius (TU Wien).
Monday, June 12, 15:30 ~ 16:00
A posteriori error estimates for singularly perturbed equations
Natalia Kopteva
University of Limerick, Ireland - This email address is being protected from spambots. You need JavaScript enabled to view it.
Solutions of singularly perturbed partial differential equations typically exhibit sharp boundary and interior layers, as well as corner singularities. To obtain reliable numerical approximations of such solutions in an efficient way, one may want to use meshes that are adapted to solution singularities using a posteriori error estimates. In this talk, we shall discuss residual-type a posteriori error estimates singularly perturbed reaction-diffusion equations and singularly perturbed convection-diffusion equations. The error constants in the considered estimates are independent of the diameters of mesh elements and of the small perturbation parameter. Some earlier results will be briefly reviewed, with the main focus on the recent articles [1, 2] and more recent developments.
References:
[1] N. Kopteva, R. Rankin, Pointwise a posteriori error estimates for discontinuous Galerkin methods for singularly perturbed reaction-diffusion equations, May 2022.
[2] A. Demlow, S. Franz and N. Kopteva, Maximum norm a posteriori error estimates for convection-diffusion problems, IMA J. Numer. Anal., (2023).
Joint work with Alan Demlow (Texas A&M, USA), S. Franz (TU Dresden, Germany) and R. Rankin (University of Nottingham Ningbo, China).
Monday, June 12, 16:30 ~ 17:00
Approximation classes for adaptive time-stepping finite element methods
Cornelia Schneider
Friedrich Alexander University Erlangen, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study approximation classes for adaptive time-stepping finite element methods for time-dependent Partial Differential Equations (PDEs). We measure the approximation error in $L_2([0,T)\times\Omega)$ and consider the approximation with discontinuous finite elements in time and continuous finite elements in space, of any degree. As a byproduct we define Besov spaces for vector-valued functions on an interval and derive some embeddings, as well as Jackson- and Whitney-type estimates.
Joint work with Marcelo Actis (Universidad del Litoral, Santa Fe, Argentina) and Pedro Morin (Universidad del Litoral, Santa Fe, Argentina).
Monday, June 12, 17:00 ~ 17:30
Adaptive Mesh Refinement for arbitrary initial Triangulations
Johannes Storn
Bielefeld University, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
This talk introduces a simple initialization of the Maubach/Traxler bisection routine for adaptive mesh refinements. This initialization applies to any conforming initial triangulation. It preserves shape-regularity, satisfies the closure estimate needed for optimal convergence of adaptive schemes, and allows for the intrinsic use of existing implementations.
Joint work with Lars Diening (Bielefeld University, Germany) and Lukas Gehring (Friedrich-Schiller-Universität Jena, Germany).
Monday, June 12, 17:30 ~ 18:00
Applications of a space-time FOSLS formulation for parabolic PDEs
Gregor Gantner
Inria Paris, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
While the common space-time variational formulation of a parabolic equation results in a bilinear form that is non-coercive, [Führer, Karkulik, Space–time least-squares finite elements for parabolic equations, CAMWA (2021)] recently proved well-posedness of a space-time first-order system least-squares (FOSLS) formulation of the heat equation, which corresponds to a symmetric and coercive bilinear form. In particular, the Galerkin approximation from any conforming trial space exists and is a quasi-optimal approximation. Additionally, the least-squares functional automatically provides a reliable and efficient error estimator.
In [Gantner, Stevenson, Further results on a space-time FOSLS formulation of parabolic PDEs, M2AN (2021)], we have generalized this least-squares method to general second-order parabolic PDEs with possibly inhomogenoeus Dirichlet or Neumann boundary conditions. For homogeneous Dirichlet conditions, we present convergence of adaptive finite element methods driven by the built-in least-squares estimator. Moreover, we employ the space-time least-squares method for parameter-dependent problems as well as optimal control problems [Gantner, Stevenson, Applications of a space-time FOSLS formulation for parabolic PDEs, IMAJNA (2023)].
Joint work with Rob Stevenson (University of Amsterdam, Netherlands).
Tuesday, June 13, 14:00 ~ 14:30
Adaptive FEM for linear elliptic PDEs: Optimal complexity
Dirk Praetorius
TU Wien, Austria - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider a general nonsymmetric second-order linear elliptic PDE in the framework of the Lax–Milgram lemma. We formulate and analyze an AFEM algorithm that steers the adaptive mesh-refinement and the inexact iterative solution of the arising linear systems. More precisely, the iterative solver employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the system and, as an inner loop, a uniformly contractive algebraic solver, e.g., an optimally preconditioned conjugate gradient method or an optimal geometric multigrid algorithm. We show that the proposed inexact adaptive iteratively symmetrized finite element method (AISFEM) leads to full linear convergence and, for sufficiently small adaptivity parameters, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time.
The talk is based on our recent preprints "Adaptive FEM with quasi-optimal overall cost for nonsymmetric linear elliptic PDEs" (arXiv:2212.00353) and "hp-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs" (arXiv:2210.10415).
Joint work with Maximilian Brunner (TU Wien, Austria), Pascal Heid (TU München, Germany), Michael Innerberger (HHMI Janelia Research Campus, USA), Ani Miraci (TU Wien, Austria) and Julian Streitberger (TU Wien, Austria).
Tuesday, June 13, 14:30 ~ 15:00
A priori and a posteriori error analysis in ${\boldsymbol H}(\mathrm{curl})$: localization, minimal regularity, and $p$-optimality
Martin Vohralík
Inria Paris, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
We design a stable local commuting projector from the entire infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ onto its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh. The projector is defined by simple piecewise polynomial projections and is stable in the $L_2$ norm, up to data oscillation. It in particular allows to establish the equivalence of local-best and global-best approximations in ${\boldsymbol H}(\mathrm{curl})$. This in turn yields to a priori error estimates under minimal Sobolev regularity in ${\boldsymbol H}(\mathrm{curl})$, localized elementwise, optimal both in the mesh size $h$ and in the polynomial degree $p$. In the heart of the projector, there is an ${\boldsymbol H}(\mathrm{curl})$-conforming flux reconstruction procedure. This itself leads to guaranteed, fully computable, constant-free, and $p$-robust a posteriori error estimates in ${\boldsymbol H}(\mathrm{curl})$. Details can be found in [1−3].
[1] Chaumont-Frelet, Théophile and Vohralík, Martin. Equivalence of local-best and global-best approximations in ${\boldsymbol H}(\mathrm{curl})$. Calcolo 58 (2021), 53.
[2] Chaumont-Frelet, Théophile and Vohralík, Martin. $p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl−curl problem. SIAM Journal on Numerical Analysis (2023), accepted for publication.
[3] Chaumont-Frelet, Théophile and Vohralík, Martin. A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$. HAL Preprint 03817302, submitted for publication, 2022.
Joint work with Théophile Chaumont-Frelet (Inria Sophia-Antipolis).
Tuesday, June 13, 15:00 ~ 16:00
Adaptive approximation for Finite Element Methods
Peter Binev
University of South Carolina, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The talk considers the development of the ideas for establishing near-optimal convergence rates for adaptive finite element methods emphasizing on the criteria for refining the adaptive meshes and the distribution of the degrees of freedom over the domain.
Tuesday, June 13, 16:30 ~ 17:00
Convergent Two-Scale Methods for the Normalized Infinity Laplacian
Abner J Salgado
University of Tennessee, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so–called two–scale methods. We show that this method is convergent, and prove rates of convergence. These rates depend not only on the regularity of the solution, but also on whether or not the right hand side vanishes. Some extensions to this approach, like obstacle problems and symmetric Finsler norms are also considered.
Joint work with Wenbo Li (LSEC, Chinese Academy of Sciences).
Tuesday, June 13, 17:00 ~ 17:30
Multilevel norms in $H^{-s}$
Thomas Führer
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we present basic ideas of the stability proof of multilevel decompositions of piecewise polynomial spaces in negative order Sobolev spaces. We consider sequences of uniform simplicial meshes as well as sequences of adaptively generated meshes. The latter requires a local projection operator in Sobolev spaces of negative order. We discuss its construction together with basic properties and possible applications. We also define multilevel norms that are equivalent to the canonical norms. Applications include the definition of efficient preconditioners for, e.g., boundary integral equations, or residual minimization in fractional order Sobolev spaces.
Tuesday, June 13, 17:30 ~ 18:00
Adaptive methods with $C^1$ splines for multi-patch surfaces
Carlotta Giannelli
University of Florence, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
We propose an adaptive isogeometric method for the numerical approximation of (high order) partial differential equations defined on multi-patch surfaces. By focusing on $C^1$ hierarchical spline constructions, we will present a refinement algorithm with linear complexity which guarantees the construction of suitably graded hierarchical meshes that fulfill the condition for linear independence of the hierarchical basis. A selection of numerical examples will confirm the potential of the adaptive scheme on different multipatch configurations.
Joint work with Cesare Bracco (University of Florence, Italy), Andrea Farahat (RICAM, Austria), Mario Kapl (Carinthia University of Applied Sciences, Villach, Austria), Rafael Vázquez (EPFL, Lausanne, Switzerland).
Wednesday, June 14, 14:00 ~ 14:30
Inf-Sup Theory for the Biot equations. Part 1: Analysis
Christian Kreuzer
TU Dortmund University, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We present a new analysis for the quasi-static Biot equations, which modells the flow of a Newtonian fluid inside an elastic porous medium. The main unknowns are the displacement of the elastic medium and the pressure of the fluid. The presented analysis is based on the Banach-Necas Theorem and thus implies existence of a unique solution for data with minimal regularity. Moreover, the resulting variational setting may guide the design and analysis of quasi-optimal finite element methods, which will be presented in another talk by Pietro Zanotti.
After an introduction to the Biot problem, we shall discuss the existing existence and uniqueness analysis for the Biot problem and their limitations. In particular, we consider the required data regularity and compare it with the naively expected regularity. From the gained insight, we will fix a norm for the `naive' test-spaces and prove inf-sup stability of the bilinear form thereby fixing the norm on the trial spaces. It turns out that the controlled trail norm defines a slightly bigger trial space compared to previous results; an example shows that the inclusion is strict.
Last but not least, we shall present a regularity `shift' theorem for certain parameter configurations.
Joint work with Pietro Zanotti (Università degli Studi di Pavia, Italy).
Wednesday, June 14, 14:30 ~ 15:00
Multiresolution Super-Localized Orthogonal Decomposition
Daniel Peterseim
University of Augsburg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce a novel multiresolution super-localized orthogonal decomposition (SLOD) for the approximation of elliptic partial differential operators with arbitrarily rough coefficients. The method merges the concepts of (S)LOD and operator-adapted wavelets (gamblets). It computes hierarchical bases that block-diagonalize the partial differential operator and thereby decouple the discretization scales. At the same time, sparsity is enforced by a novel localization strategy that leads to a super-exponential decay of the basis functions relative to their discretization scales within the hierarchy.
Joint work with José C. Garay (University of Augsburg, Germany) and Christoph Zimmer (University of Augsburg, Germany).
Wednesday, June 14, 15:00 ~ 15:30
Inf-Sup Theory for the Biot equations. Part 2: Discretization
Pietro Zanotti
Università degli Studi di Pavia, Italy - This email address is being protected from spambots. You need JavaScript enabled to view it.
The quasi-static Biot equations in poroelasticity describe the flow of a Newtonian fluid inside an elastic porous medium. We propose a new finite element discretization of the equations, based on the inf-sup theory presented in another talk by Christian Kreuzer.
The discretization involves the two main unknowns of the equations, namely the displacement of the elastic medium and pressure of the fluid, as well as the total pressure and the total fluid content, two auxiliary variables playing a central role in our analysis. We make use of the backward Euler scheme in time and approximate all variables in space by conforming Lagrange finite elements on simplicial meshes.
We establish the well-posedness, the stability and the quasi-optimality of the discretization. All the constants involved in our analysis are robust with respect to the material parameters. We additionally discuss the preconditioning of the linear system to be solved at each time step.
Joint work with Christian Kreuzer (TU Dortmund).
Wednesday, June 14, 15:30 ~ 16:00
Linear and Nonlinear Methods for Model Reduction
Andrea Bonito
Texas A&M University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider model reduction methods for the approximation of multivariate analytic functions in the case where the functions depend on infinitely many variables but present a certain anisotropy.
The usual approach to model reduction is to construct a low dimensional linear space and define the approximation as some projection into the latter. However, it’s well-known that nonlinear methods, such as adaptive or best n- term approximations, provide improved efficiency. Alternatively, we consider a collection of linear spaces (aka a library) used to approximate locally the target function. All the linear spaces in the library are of dimension considerably smaller than the dimension required for a single space to achieve the same accuracy. While exhibiting a marginal improvement in the approximation of the multivariate functions, it circumvents the inherent exponentially increasing complexity in constructing reduced spaces as their dimension increases.
We first introduce various anisotropic model classes based on Taylor expansions and study their approximation by finite dimensional polynomial spaces described by lower sets. Then, in the framework of parametric PDEs, we present a possible strategy that can be used to build a library and provide an analysis of its performance.
Wednesday, June 14, 16:30 ~ 17:00
Mesh grading of triangulations generated by adaptive bisection
Tabea Tscherpel
TU Darmstadt, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Adaptive mesh refinement algorithms such as the 2D newest vertex bisection and its generalisations to higher dimensions by Maubach and Traxler are an essential part of adaptive finite element methods. As intended, this generates triangulations with locally varying mesh size, i.e., the meshes are graded.
The $L^2$-projection mapping to the continuous Lagrange finite element spaces is an important tool in numerical analysis. For parabolic problems it is known that its stability properties in Sobolev spaces are the key to discrete stability and quasi-optimality estimates. However, its stability properties depend on the grading of the underlying triangulation, and for this reason such grading properties are of particular interest.
Previously, only grading properties for 2D mesh refinement schemes have been obtained and in higher dimensions the corresponding results are much more challenging.
We present optimal results on the grading of families generated by the adaptive bisection algorithm by Maubach and Traxler for arbitrary dimensions. Those sharpen previous results in 2D and are the first results in higher dimensions. Furthermore, we discuss the implications on Sobolev stability of the $L^2$-projection.
Joint work with Lars Diening (Bielefeld University) and Johannes Storn (Bielefeld University).
Wednesday, June 14, 17:00 ~ 17:30
Lower bounds, elliptic reconstruction and a posteriori error control of parabolic problems
Emmanuil Georgoulis
Heriot-Watt University and National Technical University of Athens, UK and Greece - This email address is being protected from spambots. You need JavaScript enabled to view it.
A popular approach for proving a posteriori error bounds in various norms for evolution problems with partial differential equations (PDEs) uses reconstruction operators to recover conforming objects from the approximate solutions. So far, lower bounds in reconstruction-based a posteriori error estimators have been proven only for time-discrete schemes for parabolic problems; the proof of lower bounds for fully discrete schemes in reconstruction-based a posteriori error estimators has eluded.
In this work, we provide a complete framework addressing this issue for energy-type norms. We consider Backward Euler discretisations and time-discontinuous Galerkin schemes, combined with dynamically changing conforming finite element methods in space, approximating linear parabolic problems. The results presented include sharp upper and lower a posteriori error bounds. Localised versions of the lower bounds are also considered.
Joint work with Charalambos Makridakis (IACM-FORTH, University of Crete & University of Sussex).
Wednesday, June 14, 17:30 ~ 18:00
Oscillations and Differences in Triebel-Lizorkin-Morrey Spaces
Markus Weimar
JMU Wuerzburg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss new characterizations of Triebel-Lizorkin-Morrey spaces $\mathcal{E}_{u,p,q}^s$ of positive smoothness $s$ in terms of local oscillations (i.e., local polynomial bestapproximations) as well as integral means of higher order differences. This family of function spaces generalizes the well-established scale of Triebel-Lizorkin spaces $F^s_{p,q}$ which particularly contains the usual $L_p$-Sobolev spaces $H^s_p=F^s_{p,2}$ as special cases. Moreover, there are strong relations to BMO and Campanato spaces. We extend assertions due to Triebel 1992 and Yuan/Sickel/Yang 2010 for spaces on $\mathbb{R}^d$ and additionally consider their restrictions to bounded Lipschitz domains $\Omega$. Furthermore, we indicate possible applications to the regularity theory of quasi-linear elliptic PDEs.
The results to be presented are based on a recent preprint [1] in joint work with Marc Hovemann (Marburg).
[1] M.~Hovemann and M.~Weimar. Oscillations and differences in Triebel-Lizorkin-Morrey spaces. Preprint in preparation, 2023.
Joint work with Marc Hovemann (Marburg).
Posters
Discrete time analysis for domain decomposition
Arthur Arnoult
Université Sorbonne Paris Nord, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Optimized Schwarz waveform relaxation (OSWR) is a domain decomposition algorithm for solving partial differential equations on small subdomains in order to accelerate numerical resolution. This poster shows a new approach that provides new results in the convergence analysis of OSWR iterations for parabolic problems.
This new approach relies on the time discretization of the domain decomposition equations with backward Euler, in order to obtain a system of differential equations that can be analytically solved. Contrary to the classical method that choses the Robin parameter that minimizes the contraction ratio of the Fourier transform of the continuous in time solution, this method minimizes the contraction matrix norm of the discrete time solution.
This method allows to define efficient optimized Robin parameters that depend on the targeted iteration count, a property that is shared by the actual observed optimal parameters, while traditional Fourier analysis in the time direction leads to iteration independent parameters. Numerical results show that this parameter is an accurate estimation of the optimal Robin parameter, which allows to perform the smallest number of iterations possible.
Multi-scale Finite Element Method for incompressible flow in Perforated Domain
Loïc Balazi Atchy Nillama
CEA Saclay - CMAP Ecole Polytechnique, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
Multi-scale problems arise in numerous engineering fields such as reservoir engineering, flows through fractured porous media, flows in nuclear reactor cores, etc. In these media with many obstacles of various sizes, the macroscopic flow is directly influenced by local phenomena occurring at the finest scales. Thus, computing numerically such flows requires a very fine mesh to resolve all the details. Despite the continuous increase in computer resources, these are insufficient to perform classical finite element simulations with an accuracy allowing correct resolution of the finest scales of the flow. To overcome this limitation, various multi-scale methods have been developed to attempt to resolve scales below the coarse mesh scale by incorporating local computations into a global problem which is defined only on a coarse mesh. Among the many multi-scale approaches that have been proposed in the literature, we can mention the Heterogeneous Multi-scale Method (HMM) [1], the Local Orthogonal Decomposition (LOD) [2] or the Multi-scale Finite Element Method (MsFEM). In this contribution, we focus on the Multi-scale Finite Element Method.
The Multi-scale Finite Element Method uses a coarse mesh on which one defines basis functions which are no longer the classical polynomial basis functions of finite elements, but which solve fluid mechanics equations on the elements of the coarse mesh. These functions are themselves numerically approximated on a fine mesh considering all the geometric details, which gives the multi-scale aspect of this method.
Based on the work of [3, 4], we propose to develop an enriched non-conforming Multi-scale Finite Element Method (MsFEM) to solve viscous incompressible flow in heterogeneous media. Our MsFEM is in the vein of the classical non-conforming Crouzeix-Raviart finite element method with high-order weighting functions. At the theoretical level, in order to complete the work of [3], we perform a rigorous study of this MsFEM in two and three dimensions to solve the Stokes or the Oseen equations. We show the well-posedness of the discretized local problems firstly for a family of non-conforming finite elements of arbitrary order on triangles presented in [5], and secondly for a new family of finite elements that we have developed in three dimensions. In addition, we quantify the error between the MsFEM and the exact solution for the global Stokes problem in perforated domain using homogenisation theory.
At the numerical level, we implement the Multi-Scale Finite Element Methods developed, in two and three dimensions, up to the order two, in a parallel framework using PETSc in FreeFEM [6] : the basis functions being independent of each other, their approximations as well as the assembly of the macroscopic problem can be carried out in parallel. We compare the MsFEM approximations with reference solutions obtained by performing the simulations on a fine grid with a classical Finite Element Method. Besides, we compare the Galerkin and the Petrov-Galerkin approaches for solving the Oseen equations.
The perspective of this work is now to develop a methodology to solve the Navier-Stokes equations with multi-scale basis functions. In parallel, to complete the study of our MsFEM for the Stokes equations, we are investigating on an a posteriori error estimate.
[1] W. E, B. Engquist, Z. Huang. Heterogeneous multiscale method : A general methodology for multiscale modeling. Phys. Rev. B,67, 092101, 2003. doi :10.1103/PhysRevB. 67.092101.
[2] R. Altmann, P. Henning, D. Peterseim. Numerical homogenization beyond scale separation. Acta Numerica, 30, 1–86, 2021. doi : 10.1017/S0962492921000015.
[3] Q. Feng, G. Allaire, P. Omnes. Enriched Nonconforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media Based on High-order Weighting Functions. Multiscale Modeling & Simulation, pp. 462–492, 2022. doi :10.1137/21M141926X. Publisher : Society for Industrial and Applied Mathematics.
[4] G. Jankowiak, A. Lozinski. Non-Conforming Multiscale Finite Element Method for Stokes Flowsin Heterogeneous Media. Part II : error estimates for periodic microstructure. arXiv :1802.04389[math], 2018. ArXiv : 1802.04389.
[5] G. Matthies, L. Tobiska. Inf-sup stable non-conforming finite elements of arbitrary order on triangles. Numerische Mathematik, 102, 293–309, 2005. doi : 10.1007/s00211-005-0648-8
[6] F. Hecht. New development in freefem++. J. Numer. Math.,20(3-4), 251–265, 2012.
Joint work with Grégoire Allaire (CMAP, Ecole Polytechnique, France) and Pascal Omnes (CEA Saclay, France).
A phase-space discontinuous Galerkin approximation for the radiative transfer equation in slab geometry
Riccardo Bardin
University of Twente, The Netherlands - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we consider the numerical solution of the second-order form of the radiative transfer equation (RTE) in slab geometry, employing a SIP discontinuous Galerkin method and several error estimators for adaptivity of the mesh.
Due to the product structure of the phase-space in this formulation, classical numerical schemes for the RTE, such as the $P_N$-approximations, use separate discrtization techniques for the spatial and angular variables. However, a major drawback of the independent discretizations is that a local refinement in phase-space is not possible. Therefore, the aim is to develop a numerical method for the approximation of the RTE that allows for local mesh refinement in phase-space and that allows for a relatively simple analysis and implementation [1]. To accomplish this, we employ quad-tree grids as partitions for the phase-space and a symmetric interior penalty discontinuous Galerkin formulation for the discrete problem.
Besides the proper treatment of traces, which requires the inclusion of a weight function in our case, the analysis of the overall scheme is along the standard steps for the analysis of discontinuous Galerkin methods. Supporting examples show the accuracy and stability of the method also numerically for different polynomial degrees.
For the local adaptation of the grid we investigate several error estimators. First, we consider two hierarchical error estimators, which use polynomials of higher degree, and the discrete solution on a uniformly refined mesh, respectively. To overcome the fact that these estimators require the solution of an additional global problem in every step, we then propose an a posteriori estimator based on a local averaging procedure, which shows performances comparable to the more costly hierarchical estimators.
[1] - Bardin, R., Bertrand, F., Palii, O. and Schlottbom, M. - A phase-space discontinuous Galerkin approximation for the radiative transfer equation in slab geometry, arXiv:2201.06104v3 (2023)
Joint work with Matthias Schlottbom (University of Twente, The Netherlands), Olena Palii (University of Twente, The Netherlands) and Fleurianne Bertrand (University of Twente, The Netherlands).
High-order Implicit-Explicit Time Integration for the Kinetic Simulation of Magnetized Plasmas
Debojyoti Ghosh
Lawrence Livermore National Laboratory, United States of America - This email address is being protected from spambots. You need JavaScript enabled to view it.
Kinetic models for the plasma dynamics in the edge region of a tokamak (magnetic confinement fusion device) are characterized by a wide range of time scales that span several orders of magnitude. These may include ion and electron transport time scales, acoustic waves, collisional processes, Alfven waves, and the plasma frequency. This makes efficient numerical time integration challenging, where the scales of interest are slower than the fastest scales in the model. We describe the implementation and performance of high-order implicit-explicit (IMEX) methods in COGENT, a finite-volume, open-source code for gyrokinetic and fluid simulations of magnetized plasmas in complex geometries [1]. The governing equations include an arbitrary number of kinetic or fluid models of charged species and may include fluid models for neutral species. The kinetic partial differential equations (PDEs) are discretized on high-dimensional "phase-space" (physical and velocity space) grids, while the fluid PDEs are discretized on physical space grids; thus, the algorithm evolves solutions with multiple dimensionalities. The tokamak-edge geometry is represented with mapped, multiblock grids, and spatial derivatives are discretized with a fourth-order finite-volume method in the mapped coordinates [2]. We implement high-order multi-stage additive Runge-Kutta (ARK) methods [3] that require the partitioning of the right-hand-side (RHS) of the semi-discrete ordinary differential equation (ODE) into “stiff” and “nonstiff” components; the stiff component is integrated implicitly in time, while the nonstiff component is integrated explicitly. We present a modified ARK method that allows a nonlinear function on the left-hand-side, i.e., it solves an ODE of the form $d[M(u)]/dt = R(u)$. The resulting nonlinear system is solved using the Jacobian-free Newton Krylov (JFNK) approach [4]. The performance of the IMEX time integration depends on effectively preconditioning the linear solve; we describe an operator-split multiphysics preconditioner where tailored preconditioners for each implicit physics term are wrapped in an operator-split algorithm to precondition the complete stiff RHS. We investigate the computational performance and convergence of the ARK methods for simulations representative of tokamak-edge plasma dynamics.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.
[1] COGENT, https://github.com/LLNL/COGENT
[2] Dorr, M., Colella, P., Dorf., M., Ghosh, D., Hittinger, J., Schwartz, P. O., High-order Discretization of a Gyrokinetic Vlasov Model in Edge Plasma Geometry, Journal of Computational Physics, 373, 2018, 605-630.
[3] Kennedy, C. A., Carpenter, M. H., Additive Runge–Kutta schemes for convection–diffusion–reaction equations, Applied Numerical Mathematics, 44, 2003, 139-181.
[4] Knoll, D. A., Keyes, D. E., Jacobian-free Newton–Krylov methods: a survey of approaches and applications, Journal of Computational Physics, 193, 2004, 357-397.
Joint work with Milo Dorr (Lawrence Livermore National Laboratory), Mikhail Dorf (Lawrence Livermore National Laboratory) and Lee Ricketson (Lawrence Livermore National Laboratory).
Approximation of Functionals by Neural Network without Curse of Dimensionality
Tianyu Jin
The Hong Kong University of Science and Technology, Hong Kong - This email address is being protected from spambots. You need JavaScript enabled to view it.
Recently, many methods have been developed for solving partial differential equations (PDEs) by neural networks. However, the curse of dimensionality (CoD) is a serious issue that generally exists in this field when dealing with high dimensional problems. In this work, we establish a new method for the approximation of functionals by neural networks without CoD by defining (i) a Fourier-type series on the infinite-dimensional space of functionals and (ii) the associated Barron spectral space $\mathcal{B}_s$ and a Hilbert space $\mathcal{H}_s$ of functionals. The approximation error of the designed neural network in this method is $O(1/\sqrt{m})$ where $m$ is the size of networks. Then, the proposed method is employed in several numerical experiments, such as evaluating the energy functionals, solving two-dimensional and four-dimensional Poisson equations by aforementioned neural networks at one or a few given points.
Joint work with YANG Yahong (The Hong Kong University of Science and Technology, Hong Kong) and XIANG Yang (The Hong Kong University of Science and Technology, Hong Kong).
Node Subsampling for \\Multilevel Meshfree Elliptic PDE Solvers
Andrew Lawrence
University of Colorado - Boulder, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Subsampling of node sets is useful in contexts such as multilevel methods, polynomial approximation, and numerical integration. On uniform grid-based node sets, the process of subsampling is simple. However, on non-uniform node sets, the process of coarsening a node set through node elimination is nontrivial. A novel method for such subsampling is presented here. Additionally, boundary preservation techniques and two novel node set quality measures are presented. The new subsampling method is demonstrated on the test problems of solving the Poisson and Laplace equations by multilevel radial basis function-generated finite differences (RBF-FD) iterations.
Joint work with Morten E. Nielsen and Bengt Fornberg.
A posteriori error analysis of a linear Schrödinger type eigenvalue problem for atomic centered discretizations
Ioanna-Maria Lygatsika
Sorbonne Université, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster, we present a first a posteriori error analysis for variational approximations of the ground state eigenpair of a linear Schrödinger type eigenvalue problem for systems with one electron and $M$ atoms, more precisely of the form $Hu=\lambda u$, $H=-\Delta + \sum_{i=1}^M V_i + \sigma$, $\|u\|_{L^2}=1$, with boundary conditions in one dimension. Denoting by $(u_N,\lambda_N)$ the variational approximation of the ground state eigenpair $(u,\lambda)$ based on a Gaussian discretization centered on atoms, we provide a posteriori estimates of the error in the energy norm $\|u - u_N\|_H$, when $N$ goes to infinity. We introduce the residual of the equation and we decompose it into $M$ residuals characterizing the error localized on atoms. It is shown that the bound can be expressed in terms of the dual "local" norms induced by the radially symmetric operators $H_i=-\Delta + V_i + \sigma_i, i=1,\ldots,M$ centered on atoms. Such bound is fully computable as soon as an estimate on the dual local norms of the local residuals is available, which is obtained by performing a spectral decomposition of the bounded operators $H_i$ of Hydrogen-like atoms. Finally, we provide numerical illustration of the performance of such a posteriori analysis on test cases.
Joint work with Mi-Song Dupuy (Sorbonne Université, France) and Geneviève Dusson (Université Bourgogne Franche-Comté, France).
A physics-inspired neural network combined with a library-search algorithm in inverse problems of Schrödinger equation
Yiran Wang
Purdue University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we solve inverse problems of a Schrödinger equation that can be formulated as a learning process of a convolutional neural network combined with a special projection technique which is called a library-search algorithm. The Schrödinger equation is $i\frac{\partial \psi}{\partial t}-\beta\frac{\partial^2 \psi}{\partial x^2}+\gamma|\psi|^2\psi+V(x)\psi=0,$ where the wave function $\psi(x,t)$ is the solution to the forward problem and the potential $V(x)$ is the quantity of interest of the inverse problem. The main contributions of this work come from two aspects. First, we construct a special neural network directly from the Schrödinger equation, which is motivated by a splitting method. The physics behind the construction enhances the explainability of the neural network. In particular, each convolution layer and activation function correspond to different parts of the equation, which can be a useful guild when we train the network. Under this construction, it can be rigorously proved that the neural network has a convergence rate with respect to the length of input data and number of layers. The other part is using a library-search algorithm to project the solution space of the inverse problem to a lower-dimensional space. The motivation of this part is to alleviate the training burden in estimating functions. Instead, with a well-chosen library, we can greatly simplify the training process. More specifically, in one of the experiments, we analysis the landscape of the loss function with respect to the training parameters to easily obtain the optimal solution to the inverse problem. A brief version of analysis is given, which is focused on the well-possedness of some mentioned inverse problems and convergence of the neural network approximation. To show the effectiveness of the proposed method, we explore in some representative problems including simple equations and a couple equation. The results can well verify the theory part. In the future, we can further explore manifold learning to enhance the approximation effect of the library-search algorithm.
Rate-Optimal Sparse Approximation of Compact Break-of-Scale Embeddings
Markus Weimar
JMU Wuerzburg, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
The poster addresses the approximation problem of functions in new scales of function spaces with hybrid smoothness. In these scales we combine classical (isotropic) regularity measured in $L_p$ with so-called dominating mixed smoothness which arises in high-dimemsional real-world applications, e.g., related to the electronic Schrödinger equation. Sharp dimension-independent rates of convergence for linear and nonlinear best approximations using $n$ hyperbolic wavelets are presented. Important special cases include the approximation of function having dominating mixed smoothness w.r.t. $L_p$ in the norm of the isotropic energy space $H^1$.
The presented results are based on a recent paper [1] which represents the first part of a long term research project.
[1] G. Byrenheid, J. Hübner, and M. Weimar. Rate-optimal sparse approximation of compact break-of-scale embeddings. Appl. Comput. Harmon. Anal. 65:40--66, 2023 (arXiv:2203.10011).
Joint work with Glenn Byrenheid (FSU Jena, Germany), Janina Hübner (RUB, Germany) and Markus Hansen (PU Marburg, Germany).