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A POSTERIORI ERROR ESTIMATION AND ADAPTIVITY FOR MULTIPLE-NETWORK POROELASTICITY

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arXiv 2021年

作者： Eliseussen, Emilie Rognes, Marie E. Thompson, Travis B. Institute of Basal Medical Sciences University of Oslo Oslo Norway Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Department of Mathematics University of Bergen Bergen Norway Department of Mathematics and Statistics Texas Tech University LubbockTX United States

The multiple-network poroelasticity (MPET) equations describe deformation and pressures in an elastic medium permeated by interacting fluid networks. In this paper, we (i) place these equations in the theoretical context of coupled elliptic-parabolic problems, (ii) use this context to derive residual-based a posteriori error estimates and indicators for fully discrete MPET solutions and (iii) evaluate the performance of these error estimators in adaptive algorithms for a set of test cases: ranging from synthetic scenarios to physiologically realistic simulations of brain *** Codes 65M60 (Primary), 92B05 (Secondary) © 2021, CC BY.

关键词： Adaptive algorithms

On reference solutions and the sensitivity of the 2D Kelvin–Helmholtz instability problem

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arXiv 2018年

作者： Schroeder, Philipp W. John, Volker Lederer, Philip L. Lehrenfeld, Christoph Lube, Gert Schöberl, Joachim Institute for Numerical and Applied Mathematics Georg-August-Universität Göttingen Göttingen37083 Germany Weierstrass Institute for Applied Analysis and Stochastics Berlin10117 Germany Department of Mathematics and Computer Science Freie Universität Berlin Berlin14195 Germany Institute for Analysis and Scientific Computing TU Wien 1040 Wien Austria

Two-dimensional Kelvin–Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin–Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed. Copyright © 2018, The Authors. All rights reserved.

关键词： numerical methods

A PARALLEL RANK-ADAPTIVE INTEGRATOR FOR DYNAMICAL LOW-RANK APPROXIMATION

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arXiv 2023年

作者： Ceruti, Gianluca Kusch, Jonas Lubich, Christian Institute of Mathematics EPF Lausanne Lausanne1015 Switzerland Numerical Analysis and Scientific Computing Universität Innsbruck Technikerstraße 13 Innsbruck6020 Austria Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 Tübingen72076 Germany

This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update & Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy. Copyright © 2023, The Authors. All rights reserved.

关键词： Differential equations

A DISCONTINUOUS GALERKIN METHOD FOR THE EXTRACELLULAR MEMBRANE INTRACELLULAR MODEL

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arXiv 2024年

作者： Masri, Rami Kirk, Keegan L.A. Hauge, Eirill Kuchta, Miroslav Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway Division of Applied Mathematics Brown University ProvidenceRI02906 United States Department of Mathematical Sciences Center for Mathematics and Artificial Intelligence George Mason University FairfaxVA22030 United States

We formulate and analyze interior penalty discontinuous Galerkin methods for coupled elliptic PDEs modeling excitable tissue, represented by intracellular and extracellular domains sharing a common interface. The PDEs are coupled through a dynamic boundary condition, posed on the interface, that relates the normal gradients of the solutions to the time derivative of their jump. This system is referred to as the Extracellular Membrane Intracellular model or the cell-by-cell model. Due to the dynamic nature of the interface condition and to the presence of corner singularities, the analysis of discontinuous Galerkin methods is non-standard. We prove the existence and uniqueness of solutions by a reformulation of the problem to one posed on the membrane. Convergence is shown by utilizing face-to-element lifting operators and notions of weak consistency suitable for solutions with low spatial regularity. Further, we present parameter-robust preconditioned iterative solvers. numerical examples in idealized geometries demonstrate our theoretical findings, and simulations in multiple cells portray the robustness of the *** Codes 65N30, 65M60 Copyright © 2024, The Authors. All rights reserved.

关键词： Galerkin methods

Brain solute transport is more rapid in periarterial than perivenous spaces

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Research Square

Research Square 2021年

作者： Vinje, Vegard Bakker, Erik Nicolaas Theodorus Petrus Rognes, Marie E. Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Martin Linges vei 25 Fornebu 1364 Norway Amsterdam University Medical Center Dept. of Biomedical Engineering and Physics Meibergdreef 9 Amsterdam1105 AZ Netherlands University of Bergen Department of Mathematics Bergen Norway

Background: Perivascular fluid flow, of cerebrospinal or interstitial fluid in spaces surrounding brain blood vessels, is recognized as a key component underlying brain transport and clearance. An important open question is how and to what extent differences in vessel type or geometry affect perivascular fluid flow and transport. Methods: Using computational modelling in both idealized and image-based geometries, we study and compare fluid flow and solute transport in pial (surface) periarterial and perivenous spaces. Results: Our findings demonstrate that differences in geometry between arterial and venous pial perivascular spaces (PVSs) lead to higher net CSF flow, more rapid tracer transport and earlier arrival times of injected tracers in periarterial spaces compared to perivenous spaces. Conclusions: These findings can explain the experimentally observed rapid appearance of tracers around arteries, and the delayed appearance around veins without the need of a circulation through the parenchyma, but rather by direct transport along the PVSs. © 2021, CC BY.

关键词： Blood vessels

Efficient implementation of adaptive P1-FEM in Matlab

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Computational Methods in Applied Mathematics 2011年第4期11卷 460-490页

作者： Funken, Stefan Praetorius, Dirk Wissgott, Philipp Institute for Numerical Mathematics University of Ulm Helmholtzstraße 18 D-89069 Ulm Germany Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstraße 8-10 A-1040 Wien Austria Institute for Solid State Physics Vienna University of Technology Wiedner Hauptstraße 8-10 A-1040 Wien Austria

We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive mesh-refining algorithm. Throughout, the focus is on an efficient realization by use of MATLAB built-in functions and vectorization. numerical experiments underline the efficiency of the code which is observed to be of almost linear complexity with respect to the runtime. Although the scope of this paper is on AFEM, the general ideas can be understood as a guideline for writing efficient MATLAB code. © 2011 Institute of Mathematics, National Academy of Sciences.

关键词： MATLAB

A polynomial-degree-robust a posteriori error estimator for nédélec discretizations of magnetostatic problems

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arXiv 2020年

作者： Gedicke, Joscha Geevers, Sjoerd Perugia, Ilaria Schöberl, Joachim Institute for Numerical Simulation University of Bonn Endenicher Allee 19b Bonn53115 Germany Faculty of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 Vienna1090 Austria Institute for Analysis and Scientific Computing Vienna University of Technology Wiedner Hauptstrasse 8-10 Vienna1040 Austria

We present an equilibration-based a posteriori error estimator for Nédélec element discretizations of the magnetostatic problem. The estimator is obtained by adding a gradient correction to the estimator for Nédélec elements of arbitrary degree presented in [16]. This new estimator is proven to be reliable, with reliability constant 1, and efficient, with an efficiency constant that is independent of the polynomial degree of the approximation. These properties are demonstrated in a series of numerical experiments on three-dimensional test *** Codes 65N15, 65N30, 65N50 Copyright © 2020, The Authors. All rights reserved.

关键词： Magnetostatics

ON THE IMPROVED CONVERGENCE OF LIFTED DISTRIBUTIONAL GAUSS CURVATURE FROM REGGE ELEMENTS

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arXiv 2024年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2024, The Authors. All rights reserved.

关键词： Tensors

GENERALIZING RIEMANN CURVATURE TO REGGE METRICS

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arXiv 2023年

作者： Gopalakrishnan, Jay Neunteufel, Michael Schöberl, Joachim Wardetzky, Max Portland State University PO Box 751 PortlandOR97201 United States Institute for Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 Wien1040 Austria Institute of Numerical and Applied Mathematics University of Göttingen Lotzestr. 16-18 Göttingen37083 Germany

In this paper we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components are assumed. While linear derivatives of the metric can be generalized as Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization combining the classical angle defect and jumps of the second fundamental form across element interfaces, and rigorously prove correctness of this generalization. Specifically, if a piecewise smooth metric approximates a globally smooth metric, our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor arising from a globally smooth metric. Moreover, we show that if the metric approximation converges at some rate in a piecewise norm that scales like the L2-norm, then the curvature approximation converges in the H-2-norm at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any *** Codes 65N30 (Primary) 53A70, 83C27 (Secondary) Copyright © 2023, The Authors. All rights reserved.

关键词： Tensors