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New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces 利用Banach空间求解Navier-Stokes-Brinkman方程的非增广混合有限元新方法
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2023-03-24 DOI: 10.1515/jnma-2022-0073
G. Gatica, Nicolás Núñez, R. Ruiz-Baier
{"title":"New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces","authors":"G. Gatica, Nicolás Núñez, R. Ruiz-Baier","doi":"10.1515/jnma-2022-0073","DOIUrl":"https://doi.org/10.1515/jnma-2022-0073","url":null,"abstract":"Abstract In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87723083","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}
引用次数: 4
Exploring numerical blow-up phenomena for the Keller–Segel–Navier–Stokes equations 探索Keller-Segel-Navier-Stokes方程的数值爆破现象
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2023-01-31 DOI: 10.48550/arXiv.2302.00139
J. Bonilla, J. V. Guti'errez-Santacreu
{"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":null,"pages":null},"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}
引用次数: 0
Unifying a posteriori error analysis of five piecewise quadratic discretisations for the biharmonic equation 统一双调和方程五个分段二次离散的后验误差分析
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2023-01-31 DOI: 10.1515/jnma-2022-0092
C. Carstensen, Benedikt Gräßle, N. Nataraj
{"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":null,"pages":null},"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}
引用次数: 0
Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems 反应扩散PDE系统图灵模式逼近的自适应POD-DEIM校正
2区 数学
Journal of Numerical Mathematics Pub Date : 2023-01-20 DOI: 10.1515/jnma-2022-0025
Alessandro Alla, Angela Monti, Ivonne Sgura
{"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":null,"pages":null},"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}
引用次数: 1
Transformed primal-dual methods for nonlinear saddle point systems 非线性鞍点系统的变换原对偶方法
2区 数学
Journal of Numerical Mathematics Pub Date : 2023-01-15 DOI: 10.1515/jnma-2022-0056
Long Chen, Jingrong Wei
{"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":null,"pages":null},"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}
引用次数: 1
Diagonally implicit Runge-Kutta schemes: Discrete energy-balance laws and compactness properties 对角隐式龙格-库塔格式:离散能量平衡定律和紧性
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2022-12-24 DOI: 10.1515/jnma-2022-0069
Abner J. Salgado, Ignacio Tomas
{"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":null,"pages":null},"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}
引用次数: 0
Error analysis for a Crouzeix–Raviart approximation of the p-Dirichlet problem p-Dirichlet问题的Crouzeix-Raviart近似的误差分析
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2022-10-21 DOI: 10.48550/arXiv.2210.12116
A. Kaltenbach
{"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":null,"pages":null},"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}
引用次数: 4
A subspace of linear nonconforming finite element for nearly incompressible elasticity and Stokes flow 近似不可压缩弹性和Stokes流的线性非协调有限元子空间
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2022-09-14 DOI: 10.1515/jnma-2022-0010
Shangyou Zhang
{"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":null,"pages":null},"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}
引用次数: 0
Transformed primal-dual methods for nonlinear saddle point systems 非线性鞍点系统的变换原对偶方法
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2022-08-04 DOI: 10.48550/arXiv.2208.02444
Long Chen, Jingrong Wei
{"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":null,"pages":null},"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}
引用次数: 4
The deal.II library, Version 9.4 这笔交易。II库，版本9.4
IF 3 2区 数学
Journal of Numerical Mathematics Pub Date : 2022-07-17 DOI: 10.1515/jnma-2022-0054
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
{"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":null,"pages":null},"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}
引用次数: 113
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