SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 251-276, February 2026. Abstract. We address the convergence analysis of lattice Boltzmann methods for scalar nonlinear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and [math]-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.
{"title":"Monotonicity and Convergence of Two-Relaxation-Times Lattice Boltzmann Schemes for a Nonlinear Conservation Law","authors":"Denise Aregba-Driollet, Thomas Bellotti","doi":"10.1137/25m1725218","DOIUrl":"https://doi.org/10.1137/25m1725218","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 251-276, February 2026. <br/> Abstract. We address the convergence analysis of lattice Boltzmann methods for scalar nonlinear conservation laws, focusing on two-relaxation-times (TRT) schemes. Unlike Finite Difference/Finite Volume methods, lattice Boltzmann schemes offer exceptional computational efficiency and parallelization capabilities. However, their monotonicity and [math]-stability remain underexplored. Extending existing results on simpler BGK schemes, we derive conditions ensuring that TRT schemes are monotone and stable by leveraging their unique relaxation structure. Our analysis culminates in proving convergence of the numerical solution to the weak entropy solution of the conservation law. Compared to BGK schemes, TRT schemes achieve reduced numerical diffusion while retaining provable convergence. Numerical experiments validate and illustrate the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146134778","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 64, Issue 1, Page 224-250, February 2026. Abstract. The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.
{"title":"A Primal-Dual Level Set Method for Computing Geodesic Distances","authors":"Hailiang Liu, Laura Zinnel","doi":"10.1137/24m1721086","DOIUrl":"https://doi.org/10.1137/24m1721086","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 224-250, February 2026. <br/> Abstract. The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"89 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146121993","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}
Ziqian Li, Kang Liu, Lorenzo Liverani, Enrique Zuazua
SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 193-223, February 2026. Abstract. In this paper, we introduce semiautonomous neural ODEs (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses, we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.
{"title":"Universal Approximation of Dynamical Systems by Semiautonomous Neural ODEs and Applications","authors":"Ziqian Li, Kang Liu, Lorenzo Liverani, Enrique Zuazua","doi":"10.1137/24m1679690","DOIUrl":"https://doi.org/10.1137/24m1679690","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 193-223, February 2026. <br/> Abstract. In this paper, we introduce semiautonomous neural ODEs (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses, we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"94 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101983","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 64, Issue 1, Page 170-192, February 2026. Abstract. Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies.
{"title":"Support Graph Preconditioners for Off-Lattice Cell-Based Models","authors":"Justin Steinman, Andreas Buttenschön","doi":"10.1137/25m1727904","DOIUrl":"https://doi.org/10.1137/25m1727904","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 170-192, February 2026. <br/> Abstract. Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"289 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101992","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 64, Issue 1, Page 148-169, February 2026. Abstract. In this paper, we propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere [math]. We employ quadrature rules that satisfy the Marcinkiewicz–Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral, with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical [math] error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution and the other characterized by the Marcinkiewicz–Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.
{"title":"Spherical Configurations and Quadrature Methods for Integral Equations of the Second Kind","authors":"Congpei An, Hao-Ning Wu","doi":"10.1137/24m1688370","DOIUrl":"https://doi.org/10.1137/24m1688370","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 148-169, February 2026. <br/> Abstract. In this paper, we propose and analyze a product integration method for the second-kind integral equation with weakly singular and continuous kernels on the unit sphere [math]. We employ quadrature rules that satisfy the Marcinkiewicz–Zygmund property to construct hyperinterpolation for approximating the product of the continuous kernel and the solution in terms of spherical harmonics. By leveraging this property, we significantly expand the family of candidate quadrature rules and establish a connection between the geometrical information of the quadrature points and the error analysis of the method. We then utilize product integral rules to evaluate the singular integral, with the integrand being the product of the singular kernel and each spherical harmonic. We derive a practical [math] error bound, which consists of two terms: one controlled by the best approximation of the product of the continuous kernel and the solution and the other characterized by the Marcinkiewicz–Zygmund property and the best approximation polynomial of this product. Numerical examples validate our numerical analysis.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"92 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146101984","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 64, Issue 1, Page 125-147, February 2026. Abstract. Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the [math] projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing preasymptotic subgeometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.
{"title":"Scaling Optimized Hermite Approximation Methods","authors":"Hao Hu, Haijun Yu","doi":"10.1137/25m1737146","DOIUrl":"https://doi.org/10.1137/25m1737146","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 125-147, February 2026. <br/> Abstract. Hermite polynomials and functions have extensive applications in scientific and engineering problems. Although it is recognized that employing the scaled Hermite functions rather than the standard ones can remarkably enhance the approximation performance, the understanding of the scaling factor remains insufficient. Due to the lack of theoretical analysis, recent publications still cast doubt on whether the Hermite spectral method is inferior to other methods. To dispel this doubt, we show in this article that the inefficiency of the Hermite spectral method comes from the imbalance in the decay speed of the objective function within the spatial and frequency domains. Proper scaling can render the Hermite spectral methods comparable to other methods. To make it solid, we propose a novel error analysis framework for the scaled Hermite approximation. Taking the [math] projection error as an example, our framework illustrates that there are three different components of errors: the spatial truncation error, the frequency truncation error, and the Hermite spectral approximation error. Through this perspective, finding the optimal scaling factor is equivalent to balancing the spatial and frequency truncation errors. As applications, we show that geometric convergence can be recovered by proper scaling for a class of functions. Furthermore, we show that proper scaling can double the convergence order for smooth functions with algebraic decay. The perplexing preasymptotic subgeometric convergence when approximating algebraic decay functions can be perfectly explained by this framework.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"286 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146089893","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 64, Issue 1, Page 76-102, February 2026. Abstract. The commonly used velocity discretization for simulating the radiative transport equation (RTE) is the discrete ordinates method (DOM). One of the long-standing drawbacks of DOM is the phenomenon known as the ray effect. Due to the high dimensionality of the RTE, DOM results in a large algebraic system to solve. The source iteration (SI) method is the most standard iterative method for solving this system. In this paper, by introducing randomness into the SI method, we propose a novel random source iteration (RSI) method that offers a new way to mitigate the ray effect without increasing the computational cost. We have rigorously proved that RSI is unbiased with respect to the SI method and that its variance is uniformly bounded across iteration steps; thus, the convergence order with respect to the number of samples is [math]. Furthermore, we prove that the RSI iteration process, as a Markov chain, is ergodic under mild assumptions. Numerical examples are presented to demonstrate the convergence of RSI and its effectiveness in mitigating the ray effect.
{"title":"Random Source Iteration Method: Mitigating the Ray Effect in the Discrete Ordinates Method","authors":"Jingyi Fu, Lei Li, Min Tang","doi":"10.1137/24m1669748","DOIUrl":"https://doi.org/10.1137/24m1669748","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 76-102, February 2026. <br/> Abstract. The commonly used velocity discretization for simulating the radiative transport equation (RTE) is the discrete ordinates method (DOM). One of the long-standing drawbacks of DOM is the phenomenon known as the ray effect. Due to the high dimensionality of the RTE, DOM results in a large algebraic system to solve. The source iteration (SI) method is the most standard iterative method for solving this system. In this paper, by introducing randomness into the SI method, we propose a novel random source iteration (RSI) method that offers a new way to mitigate the ray effect without increasing the computational cost. We have rigorously proved that RSI is unbiased with respect to the SI method and that its variance is uniformly bounded across iteration steps; thus, the convergence order with respect to the number of samples is [math]. Furthermore, we prove that the RSI iteration process, as a Markov chain, is ergodic under mild assumptions. Numerical examples are presented to demonstrate the convergence of RSI and its effectiveness in mitigating the ray effect.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070418","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 64, Issue 1, Page 103-124, February 2026. Abstract. We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.
{"title":"Second Order in Time Finite Element Schemes for Curve Shortening Flow and Curve Diffusion","authors":"Klaus Deckelnick, Robert Nürnberg","doi":"10.1137/25m1737523","DOIUrl":"https://doi.org/10.1137/25m1737523","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 103-124, February 2026. <br/> Abstract. We prove optimal error bounds for a second order in time finite element approximation of curve shortening flow in possibly higher codimension. In addition, we introduce a second order in time method for curve diffusion. Both schemes are based on variational formulations of strictly parabolic systems of partial differential equations that feature a tangential velocity which under discretization is beneficial for the mesh quality. In each time step only two linear systems need to be solved. Numerical experiments demonstrate second order convergence as well as asymptotic equidistribution.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070417","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 64, Issue 1, Page 55-75, February 2026. Abstract. The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree–dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.
{"title":"On the Convergence of Higher-Order Finite Element Methods for Nonlinear Magnetostatics","authors":"H. Egger, F. Engertsberger, B. Radu","doi":"10.1137/24m168814x","DOIUrl":"https://doi.org/10.1137/24m168814x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 55-75, February 2026. <br/> Abstract. The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or magneto-quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities, and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level, and a full convergence analysis of the resulting finite element schemes is presented, indicating order-optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We also prove local quadratic convergence in a mesh size and polynomial degree–dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145995151","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 64, Issue 1, Page 29-54, February 2026. Abstract. We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite element discretizations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analyzed as coarse spaces for the Helmholtz equation, as well as all spaces in which the Galerkin solutions are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.
{"title":"Convergence Theory for Two-Level Hybrid Schwarz Preconditioners for High-Frequency Helmholtz Problems","authors":"J. Galkowski, E. A. Spence","doi":"10.1137/25m1726972","DOIUrl":"https://doi.org/10.1137/25m1726972","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 64, Issue 1, Page 29-54, February 2026. <br/> Abstract. We give a novel convergence theory for two-level hybrid Schwarz domain-decomposition (DD) methods for finite element discretizations of the high-frequency Helmholtz equation. This theory gives sufficient conditions for the preconditioned matrix to be close to the identity and covers DD subdomains of arbitrary size, arbitrary absorbing layers/boundary conditions on both the global and local Helmholtz problems, and coarse spaces not necessarily related to the subdomains. The assumptions on the coarse space are satisfied by the approximation spaces using problem-adapted basis functions that have been recently analyzed as coarse spaces for the Helmholtz equation, as well as all spaces in which the Galerkin solutions are known to be quasi-optimal via a Schatz-type argument. As an example, we apply this theory when the coarse space consists of piecewise polynomials; these are then the first rigorous convergence results about a two-level Schwarz preconditioner applied to the high-frequency Helmholtz equation with a coarse space that does not consist of problem-adapted basis functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145919768","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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