Paola F Antonietti, Michele Botti, Ilario Mazzieri
This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.
{"title":"Discontinuous Galerkin discretization of coupled poroelasticity–elasticity problems","authors":"Paola F Antonietti, Michele Botti, Ilario Mazzieri","doi":"10.1093/imanum/drae093","DOIUrl":"https://doi.org/10.1093/imanum/drae093","url":null,"abstract":"This work is concerned with the analysis of a space–time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic–elastic media. The mathematical model consists of the low-frequency Biot’s equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is coupled with a dG time integration scheme, resulting in a full space–time dG discretization. We present the stability analysis for both semidiscrete and fully discrete formulations, and derive error estimates in suitable energy norms. The method is applied to various numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"33 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888925","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}
The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.
铁磁材料的磁化动力学由Landau-Lifshitz-Gilbert方程控制，该方程具有高度非线性，具有非凸球约束$|{textbf{m}}|=1$。设计数值格式的一个关键问题是在离散水平上保持这个球体约束。一种流行的数值方法是归一化切平面有限元法（normalized tangent plane finite element method，简称np - fem），该方法最早由Alouges和Jaisson提出，后来应用于解决各种实际问题。由于经典的能量方法不能直接应用到该方法的分析中，以往的研究只关注于收敛性，到目前为止，还没有对这种NTP-FEM进行误差估计。本文给出了严格的误差分析，并建立了最优的$H^{1}$误差估计。数值结果证实了我们的理论分析。
{"title":"Optimal error analysis of the normalized tangent plane FEM for Landau–Lifshitz–Gilbert equation","authors":"Rong An, Yonglin Li, Weiwei Sun","doi":"10.1093/imanum/drae084","DOIUrl":"https://doi.org/10.1093/imanum/drae084","url":null,"abstract":"The dynamics of the magnetization in ferromagnetic materials is governed by the Landau–Lifshitz–Gilbert equation, which is highly nonlinear with the nonconvex sphere constraint $|{textbf{m}}|=1$. A crucial issue in designing numerical schemes is to preserve this sphere constraint in the discrete level. A popular numerical method is the normalized tangent plane finite element method (NTP-FEM), which was first proposed by Alouges and Jaisson and later, applied for solving various practical problems. Since the classical energy approach fails to be applied directly to the analysis of this method, previous studies only focused on the convergence and until now, no any error estimate was established for such an NTP-FEM. This paper presents a rigorous error analysis and establishes the optimal $H^{1}$ error estimate. Numerical results are provided to confirm our theoretical analysis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888311","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 a higher-order surface finite-element penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated, which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyse the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite-element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.
{"title":"Parametric finite-element discretization of the surface Stokes equations: inf-sup stability and discretization error analysis","authors":"Hanne Hardering, Simon Praetorius","doi":"10.1093/imanum/drae080","DOIUrl":"https://doi.org/10.1093/imanum/drae080","url":null,"abstract":"We study a higher-order surface finite-element penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated, which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyse the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite-element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888926","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}
In this article, convergence and quasi-optimal rate of convergence of an Adaptive Finite Element Method is shown for the Dirichlet boundary control problem that was proposed by Chowdhury et al. (2017, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp., 86, 1103–1126). The theoretical results are illustrated by numerical experiments.
{"title":"Convergence and quasi-optimality of an AFEM for the Dirichlet boundary control problem","authors":"Arnab Pal, Thirupathi Gudi","doi":"10.1093/imanum/drae092","DOIUrl":"https://doi.org/10.1093/imanum/drae092","url":null,"abstract":"In this article, convergence and quasi-optimal rate of convergence of an Adaptive Finite Element Method is shown for the Dirichlet boundary control problem that was proposed by Chowdhury et al. (2017, Error bounds for a Dirichlet boundary control problem based on energy spaces, Math. Comp., 86, 1103–1126). The theoretical results are illustrated by numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"139 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142888312","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 the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.
{"title":"The Milstein scheme for singular SDEs with Hölder continuous drift","authors":"Máté Gerencsér, Gerald Lampl, Chengcheng Ling","doi":"10.1093/imanum/drae083","DOIUrl":"https://doi.org/10.1093/imanum/drae083","url":null,"abstract":"We study the $L^{p}$ rate of convergence of the Milstein scheme for stochastic differential equations when the drift coefficients possess only Hölder regularity. If the diffusion is elliptic and sufficiently regular, we obtain rates consistent with the additive case. The proof relies on regularization by noise techniques, particularly stochastic sewing, which in turn requires (at least asymptotically) sharp estimates on the law of the Milstein scheme, which may be of independent interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823191","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}
In this paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate transformations along radial direction to each sub-domain, we map them into eight regular sub-blocks. Subsequently, we perform numerical simulations using classical spectral methods on these regular sub-blocks. Our analysis focuses on the optimal convergence of this approach. The numerical results demonstrate the high-order accuracy of the proposed method.
{"title":"A conforming multi-domain Legendre spectral method for solving diffusive-viscous wave equations in the exterior domain with separated star-shaped obstacles","authors":"Guoqing Yao, Zicheng Wang, Zhongqing Wang","doi":"10.1093/imanum/drae085","DOIUrl":"https://doi.org/10.1093/imanum/drae085","url":null,"abstract":"In this paper, we propose a conforming multi-domain spectral method that combines mapping techniques to solve the diffusive-viscous wave equation in the exterior domain of two complex obstacles. First, we confine the exterior domain within a relatively large rectangular computational domain. Then, we decompose the rectangular domain into two sub-domains, each containing one obstacle. By applying coordinate transformations along radial direction to each sub-domain, we map them into eight regular sub-blocks. Subsequently, we perform numerical simulations using classical spectral methods on these regular sub-blocks. Our analysis focuses on the optimal convergence of this approach. The numerical results demonstrate the high-order accuracy of the proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"46 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142823307","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}
In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $sigma _{c}$. We provide explicit bounds on $sigma _{c}$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.
{"title":"Asymptotic consistency of the WSINDy algorithm in the limit of continuum data","authors":"Daniel A Messenger, David M Bortz","doi":"10.1093/imanum/drae086","DOIUrl":"https://doi.org/10.1093/imanum/drae086","url":null,"abstract":"In this work we study the asymptotic consistency of the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy) in the identification of differential equations from noisy samples of solutions. We prove that the WSINDy estimator is unconditionally asymptotically consistent for a wide class of models that includes the Navier–Stokes, Kuramoto–Sivashinsky and Sine–Gordon equations. We thus provide a mathematically rigorous explanation for the observed robustness to noise of weak-form equation learning. Conversely, we also show that, in general, the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level exceeds a critical threshold $sigma _{c}$. We provide explicit bounds on $sigma _{c}$ in the case of Gaussian white noise and we explicitly characterize the spurious terms that arise in the case of trigonometric and/or polynomial libraries. Furthermore, we show that, if the data is suitably denoised (a simple moving average filter is sufficient), then asymptotic consistency is recovered for models with locally-Lipschitz, polynomial-growth nonlinearities. Our results reveal important aspects of weak-form equation learning, which may be used to improve future algorithms. We demonstrate our findings numerically using the Lorenz system, the cubic oscillator, a viscous Burgers-growth model and a Kuramoto–Sivashinsky-type high-order PDE.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142820569","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}
A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.
{"title":"A spectral collocation method for functional and delay differential equations","authors":"Nicholas Hale","doi":"10.1093/imanum/drae079","DOIUrl":"https://doi.org/10.1093/imanum/drae079","url":null,"abstract":"A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by nonsmooth initial data. Geometric convergence in the number of degrees of freedom is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous and state-dependent delay. The framework is a natural extension of standard spectral collocation methods and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142752867","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}
In this paper, we examine a finite element approximation of the steady $p(cdot )$-Navier–Stokes equations ($p(cdot )$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law index $p(cdot )$. Numerical experiments confirm the quasi-optimality of the a priori error estimates (for the velocity) with respect to fractional regularity assumptions on the velocity vector field and the kinematic pressure.
{"title":"Error analysis for a finite element approximation of the steady p·-Navier–Stokes equations","authors":"Luigi C Berselli, Alex Kaltenbach","doi":"10.1093/imanum/drae082","DOIUrl":"https://doi.org/10.1093/imanum/drae082","url":null,"abstract":"In this paper, we examine a finite element approximation of the steady $p(cdot )$-Navier–Stokes equations ($p(cdot )$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law index $p(cdot )$. Numerical experiments confirm the quasi-optimality of the a priori error estimates (for the velocity) with respect to fractional regularity assumptions on the velocity vector field and the kinematic pressure.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142696578","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 prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^{2}$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.
{"title":"A unified framework for the error analysis of physics-informed neural networks","authors":"Marius Zeinhofer, Rami Masri, Kent–André Mardal","doi":"10.1093/imanum/drae081","DOIUrl":"https://doi.org/10.1093/imanum/drae081","url":null,"abstract":"We prove a priori and a posteriori error estimates for physics-informed neural networks (PINNs) for linear PDEs. We analyze elliptic equations in primal and mixed form, elasticity, parabolic, hyperbolic and Stokes equations, and a PDE constrained optimization problem. For the analysis, we propose an abstract framework in the common language of bilinear forms, and we show that coercivity and continuity lead to error estimates. The obtained estimates are sharp and reveal that the $L^{2}$ penalty approach for initial and boundary conditions in the PINN formulation weakens the norm of the error decay. Finally, utilizing recent advances in PINN optimization, we present numerical examples that illustrate the ability of the method to achieve accurate solutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"14 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142678475","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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