SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1929-1955, August 2024. Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order [math] and to the finite element degree [math], whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter [math]. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters [math] and [math]. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case.
{"title":"A Priori Error Estimates of a Poisson Equation with Ventcel Boundary Conditions on Curved Meshes","authors":"Fabien Caubet, Joyce Ghantous, Charles Pierre","doi":"10.1137/23m1582497","DOIUrl":"https://doi.org/10.1137/23m1582497","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1929-1955, August 2024. <br/> Abstract. In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace–Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order [math] and to the finite element degree [math], whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter [math]. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters [math] and [math]. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141909215","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024. Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]-potential and [math]), we establish an optimal second-order error bound in the [math]-norm. For low regularity potential and nonlinearity ([math]-potential and [math]), we obtain a first-order [math]-norm error bound accompanied with a uniform [math]-norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal second-order [math]-norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.
{"title":"An Explicit and Symmetric Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity","authors":"Weizhu Bao, Chushan Wang","doi":"10.1137/23m1615656","DOIUrl":"https://doi.org/10.1137/23m1615656","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024. <br/> Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]-potential and [math]), we establish an optimal second-order error bound in the [math]-norm. For low regularity potential and nonlinearity ([math]-potential and [math]), we obtain a first-order [math]-norm error bound accompanied with a uniform [math]-norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal second-order [math]-norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141899472","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024. Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods.
{"title":"Analytic and Gevrey Class Regularity for Parametric Elliptic Eigenvalue Problems and Applications","authors":"Alexey Chernov, Tùng Lê","doi":"10.1137/23m1596296","DOIUrl":"https://doi.org/10.1137/23m1596296","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1874-1900, August 2024. <br/> Abstract. We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions, where the coefficients (and hence the solution) may depend on a parameter. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyze Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique, which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients, e.g., integration by quasi–Monte Carlo methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141895229","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1814-1843, August 2024. Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the time-dependent problem, a backward Euler dG formulation is also presented and analyzed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method.
{"title":"Discontinuous Galerkin Methods for 3D–1D Systems","authors":"Rami Masri, Miroslav Kuchta, Beatrice Riviere","doi":"10.1137/23m1627390","DOIUrl":"https://doi.org/10.1137/23m1627390","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1814-1843, August 2024. <br/> Abstract. We propose and analyze discontinuous Galerkin (dG) approximations to 3D−1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the time-dependent problem, a backward Euler dG formulation is also presented and analyzed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880239","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}
Kaushik Bhattacharya, Nikola B. Kovachki, Aakila Rajan, Andrew M. Stuart, Margaret Trautner
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024. Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.
{"title":"Learning Homogenization for Elliptic Operators","authors":"Kaushik Bhattacharya, Nikola B. Kovachki, Aakila Rajan, Andrew M. Stuart, Margaret Trautner","doi":"10.1137/23m1585015","DOIUrl":"https://doi.org/10.1137/23m1585015","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024. <br/> Abstract. Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context, in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"42 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880249","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024. Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the [math] norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.
{"title":"Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for Thin-Structure Interactions","authors":"Buyang Li, Weiwei Sun, Yupei Xie, Wenshan Yu","doi":"10.1137/23m1578401","DOIUrl":"https://doi.org/10.1137/23m1578401","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024. <br/> Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the [math] norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"74 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141857639","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1759-1781, August 2024. Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrange-type basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant.
{"title":"Polynomial Interpolation of Function Averages on Interval Segments","authors":"Ludovico Bruni Bruno, Wolfgang Erb","doi":"10.1137/23m1598271","DOIUrl":"https://doi.org/10.1137/23m1598271","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1759-1781, August 2024. <br/> Abstract. Motivated by polynomial approximations of differential forms, we study analytical and numerical properties of a polynomial interpolation problem that relies on function averages over interval segments. The usage of segment data gives rise to new theoretical and practical aspects that distinguish this problem considerably from classical nodal interpolation. We will analyze fundamental mathematical properties of this problem as existence, uniqueness, and numerical conditioning of its solution. In a few selected scenarios, we will provide concrete conditions for unisolvence and explicit Lagrange-type basis systems for its representation. To study the numerical conditioning, we will provide respective concrete bounds for the Lebesgue constant.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764413","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024. Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.
{"title":"Equations with Infinite Delay: Pseudospectral Discretization for Numerical Stability and Bifurcation in an Abstract Framework","authors":"Francesca Scarabel, Rossana Vermiglio","doi":"10.1137/23m1581133","DOIUrl":"https://doi.org/10.1137/23m1581133","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1736-1758, August 2024. <br/> Abstract. We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490–505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764421","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.
{"title":"Accurately Recover Global Quasiperiodic Systems by Finite Points","authors":"Kai Jiang, Qi Zhou, Pingwen Zhang","doi":"10.1137/23m1620247","DOIUrl":"https://doi.org/10.1137/23m1620247","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024. <br/> Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"23 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141764227","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1687-1712, August 2024. Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in [math] for any [math]. We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.
{"title":"Duality-Based Error Control for the Signorini Problem","authors":"Ben S. Ashby, Tristan Pryer","doi":"10.1137/22m1534791","DOIUrl":"https://doi.org/10.1137/22m1534791","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1687-1712, August 2024. <br/> Abstract. In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in [math], for [math] in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in [math] for any [math]. We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755083","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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