Pub Date : 2023-01-31DOI: 10.48550/arXiv.2302.00139
J. Bonilla, J. V. Guti'errez-Santacreu
Abstract The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L1(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.
{"title":"Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations","authors":"J. Bonilla, J. V. Guti'errez-Santacreu","doi":"10.48550/arXiv.2302.00139","DOIUrl":"https://doi.org/10.48550/arXiv.2302.00139","url":null,"abstract":"Abstract The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L1(Ω) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"60 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75007947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residualbased a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly overpenalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2(Ω).
{"title":"Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation","authors":"C. Carstensen, Benedikt Gräßle, N. Nataraj","doi":"10.1515/jnma-2022-0092","DOIUrl":"https://doi.org/10.1515/jnma-2022-0092","url":null,"abstract":"Abstract An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residualbased a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly overpenalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2(Ω).","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"101 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83762618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction–diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e., FitzHugh–Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.
{"title":"Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems","authors":"Alessandro Alla, Angela Monti, Ivonne Sgura","doi":"10.1515/jnma-2022-0025","DOIUrl":"https://doi.org/10.1515/jnma-2022-0025","url":null,"abstract":"Abstract We investigate a suitable application of Model Order Reduction (MOR) techniques for the numerical approximation of Turing patterns, that are stationary solutions of reaction–diffusion PDE (RD-PDE) systems. We show that solutions of surrogate models built by classical Proper Orthogonal Decomposition (POD) exhibit an unstable error behaviour over the dimension of the reduced space. To overcome this drawback, first of all, we propose a POD-DEIM technique with a correction term that includes missing information in the reduced models. To improve the computational efficiency, we propose an adaptive version of this algorithm in time that accounts for the peculiar dynamics of the RD-PDE in presence of Turing instability. We show the effectiveness of the proposed methods in terms of accuracy and computational cost for a selection of RD systems, i.e., FitzHugh–Nagumo, Schnakenberg and the morphochemical DIB models, with increasing degree of nonlinearity and more structured patterns.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134916455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit-explicit and Gauss-Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex-concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.
{"title":"Transformed primal-dual methods for nonlinear saddle point systems","authors":"Long Chen, Jingrong Wei","doi":"10.1515/jnma-2022-0056","DOIUrl":"https://doi.org/10.1515/jnma-2022-0056","url":null,"abstract":"Abstract A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit-explicit and Gauss-Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex-concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"171 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135743985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.
{"title":"Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties","authors":"Abner J. Salgado, Ignacio Tomas","doi":"10.1515/jnma-2022-0069","DOIUrl":"https://doi.org/10.1515/jnma-2022-0069","url":null,"abstract":"We study diagonally implicit Runge-Kutta (DIRK) schemes when applied to abstract evolution problems that fit into the Gelfand-triple framework. We introduce novel stability notions that are well-suited to this setting and provide simple, necessary and sufficient, conditions to verify that a DIRK scheme is stable in our sense and in Bochner-type norms. We use several popular DIRK schemes in order to illustrate cases that satisfy the required structural stability properties and cases that do not. In addition, under some mild structural conditions on the problem we can guarantee compactness of families of discrete solutions with respect to time discretization.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"32 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138530262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-21DOI: 10.48550/arXiv.2210.12116
A. Kaltenbach
Abstract In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ⩾0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ⩾0, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.
{"title":"Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem","authors":"A. Kaltenbach","doi":"10.48550/arXiv.2210.12116","DOIUrl":"https://doi.org/10.48550/arXiv.2210.12116","url":null,"abstract":"Abstract In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a (p, δ)-structure for some p ∈ (1, ∞) and δ⩾0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ⩾0, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"95 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85113245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.
{"title":"A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow","authors":"Shangyou Zhang","doi":"10.1515/jnma-2022-0010","DOIUrl":"https://doi.org/10.1515/jnma-2022-0010","url":null,"abstract":"Abstract The linear nonconforming finite element, combined with constant finite element for pressure, is stable for the Stokes problem. But it does not satisfy the discrete Korn inequality. The linear conforming finite element satisfies the discrete Korn inequality, but is not stable for the Stokes problem and fails for the nearly incompressible elasticity problems. We enrich the linear conforming finite element by some nonconforming P1 bubbles, i.e., select a subspace of the linear nonconforming finite element space, so that the resulting linear nonconforming element is both stable and conforming enough to satisfy the Korn inequality, on HTC-type triangular and tetrahedral grids. Numerical tests in 2D and 3D are presented, confirming the analysis.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"58 1","pages":"157 - 173"},"PeriodicalIF":3.0,"publicationDate":"2022-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75820637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-04DOI: 10.48550/arXiv.2208.02444
Long Chen, Jingrong Wei
Abstract A transformed primal–dual (TPD) flow is developed for a class of nonlinear smooth saddle point systemThe flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit–explicit, and Gauss–Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex–concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.
{"title":"Transformed primal-dual methods for nonlinear saddle point systems","authors":"Long Chen, Jingrong Wei","doi":"10.48550/arXiv.2208.02444","DOIUrl":"https://doi.org/10.48550/arXiv.2208.02444","url":null,"abstract":"Abstract A transformed primal–dual (TPD) flow is developed for a class of nonlinear smooth saddle point systemThe flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit–explicit, and Gauss–Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex–concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"29 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74265900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Arndt, W. Bangerth, Marco Feder, M. Fehling, Rene Gassmöller, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, S. Sticko, Bruno Turcksin, David R. Wells
Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.4.
摘要本文概述了有限元库协议的新特点。II，版本9.4。
{"title":"The deal.II library, Version 9.4","authors":"D. Arndt, W. Bangerth, Marco Feder, M. Fehling, Rene Gassmöller, T. Heister, L. Heltai, M. Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, S. Sticko, Bruno Turcksin, David R. Wells","doi":"10.1515/jnma-2022-0054","DOIUrl":"https://doi.org/10.1515/jnma-2022-0054","url":null,"abstract":"Abstract This paper provides an overview of the new features of the finite element library deal.II, version 9.4.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"104 1","pages":"231 - 246"},"PeriodicalIF":3.0,"publicationDate":"2022-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88943081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from Lq(Ω) → Lq(Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.
{"title":"Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder","authors":"M. Balázsová, M. Feistauer, A. Sändig","doi":"10.1515/jnma-2021-0113","DOIUrl":"https://doi.org/10.1515/jnma-2021-0113","url":null,"abstract":"Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from Lq(Ω) → Lq(Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":"24 1","pages":"29 - 42"},"PeriodicalIF":3.0,"publicationDate":"2022-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86078351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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