SIAM Journal on Numerical Analysis最新文献

{"title":"Analysis of Complete Radiation Boundary Conditions for Maxwell’s Equations","authors":"Seungil Kim","doi":"10.1137/24m1663417","DOIUrl":"https://doi.org/10.1137/24m1663417","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1183-1208, June 2025. <br/> Abstract. We study a high order absorbing boundary condition, the so-called complete radiation boundary condition (CRBC), for a time-harmonic electromagnetic wave propagation problem in a waveguide in [math]. The CRBC has been designed for an absorbing boundary condition for simulating wave propagations governed by the Helmholtz equation based on an optimal rational approximation to the radiation condition. In this paper we develop CRBC suitable for Maxwell’s equations and show the well-posedness of Maxwell’s equations supplemented with CRBC by using a shifted electric-to-magnetic operator taking into account a separation between sources and the fictitious boundary on which CRBC is imposed. This also leads to the exponential convergence of approximate solutions satisfying CRBC with respect to the number of CRBC parameters. Numerical examples to validate the efficient performance of CRBC will be presented as well.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144176576","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}

{"title":"On the Relationship Between the Pole Condition, Absorbing Boundary Conditions, and Perfectly Matched Layers","authors":"M. Gander, A. Schädle","doi":"10.1137/24m1690916","DOIUrl":"https://doi.org/10.1137/24m1690916","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1209-1231, June 2025. <br/> Abstract. Transparent (or exact or nonreflecting) boundary conditions are essential to truncate infinite computational domains. Since transparent boundary conditions are usually nonlocal and expensive, they must be approximated. In this paper, we study such an approximation for the Helmholtz equation on an infinite strip, based on the pole condition. We show that a discretization of the pole condition can be interpreted both as a high order absorbing boundary condition and as a perfectly matched layer, two other well-known methods for approximating a transparent boundary condition. We give an error estimate which shows exponential convergence in the absence of Wood anomalies.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144176646","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}

{"title":"Numerical Approximation of Biharmonic Wave Maps into Spheres","authors":"L’ubomír Baňas, Sebastian Herr","doi":"10.1137/24m1694471","DOIUrl":"https://doi.org/10.1137/24m1694471","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1160-1182, June 2025. <br/> Abstract. We construct a structure preserving nonconforming finite element approximation scheme for the biharmonic wave maps into spheres equations. It satisfies a discrete energy law and preserves the nonconvex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points via a discrete Lagrange multiplier. This approach restricts the spatial approximation to the (nonconforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension [math]. The convergence analysis in dimensions [math] is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the nonconforming setting. Hence, we show convergence of the numerical approximation in higher dimensions by introducing additional stabilization terms in the numerical approximation. We present numerical experiments to demonstrate the performance of the proposed numerical approximation and to illustrate the regularizing effect of the bi-Laplacian, which prevents the formation of singularities.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066704","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}

{"title":"Provably Convergent Newton–Raphson Method: Theoretically Robust Recovery of Primitive Variables in Relativistic MHD","authors":"Chaoyi Cai, Jianxian Qiu, Kailiang Wu","doi":"10.1137/24m1651873","DOIUrl":"https://doi.org/10.1137/24m1651873","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1128-1159, June 2025. <br/> Abstract. A long-standing and formidable challenge faced by all conservative numerical schemes for relativistic magnetohydrodynamics (RMHD) equations is the recovery of primitive variables from conservative ones. This process involves solving highly nonlinear equations subject to physical constraints. An ideal solver should be “robust, accurate, and fast—it is at the heart of all conservative RMHD schemes,” as emphasized in [S. C. Noble et al., Astrophys. J., 641 (2006), pp. 626–637]. Despite over three decades of research, seeking efficient solvers that can provably guarantee stability and convergence remains an open problem. This paper presents the first theoretical analysis for designing a robust, physical-constraint-preserving (PCP), and provably (quadratically) convergent Newton–Raphson (NR) method for primitive variable recovery in RMHD. Our key innovation is a unified approach for the initial guess, carefully devised based on sophisticated analysis. It ensures that the resulting NR iteration consistently converges and adheres to physical constraints throughout all NR iterations. Given the extreme nonlinearity and complexity of the iterative function, the theoretical analysis is highly nontrivial and technical. We discover a pivotal inequality for delineating the convexity and concavity of the iterative function and establish general auxiliary theories to guarantee the PCP property and convergence. We also develop theories to determine a computable initial guess within a theoretical “safe” interval. Intriguingly, we find that the unique positive root of a cubic polynomial always falls within this “safe” interval. To enhance efficiency, we propose a hybrid strategy that combines this with a more cost-effective initial value. The presented PCP NR method is versatile and can be seamlessly integrated into any RMHD numerical scheme that requires the recovery of primitive variables, potentially leading to a very broad impact in this field. As an application, we incorporate it into a discontinuous Galerkin method, resulting in fully PCP schemes. Several numerical experiments, including random tests and simulations of ultrarelativistic jet and blast problems, demonstrate the notable efficiency and robustness of the PCP NR method.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"30 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144066645","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}

{"title":"A Hypocoercivity-Exploiting Stabilized Finite Element Method for Kolmogorov Equation","authors":"Zhaonan Dong, Emmanuil H. Georgoulis, Philip J. Herbert","doi":"10.1137/24m163373x","DOIUrl":"https://doi.org/10.1137/24m163373x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1105-1127, June 2025. <br/> Abstract. We propose a new stabilized finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and, crucially, is characterized by degenerate diffusion. The stabilization is constructed so that the resulting method admits a numerical hypocoercivity property, analogous to the corresponding property of the PDE problem. More specifically, the stabilization is constructed so that a spectral gap is possible in the resulting “stronger-than-energy” stabilization norm, despite the degenerate nature of the diffusion in Kolmogorov, thereby the method has a provably robust behavior as the “time” variable goes to infinity. We consider both a spatially discrete version of the stabilized finite element method and a fully discrete version, with the time discretization realized by discontinuous Galerkin timestepping. Both stability and a priori error bounds are proven in all cases. Numerical experiments verify the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143979465","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}

{"title":"Distributional Finite Element curl div Complexes and Application to Quad Curl Problems","authors":"Long Chen, Xuehai Huang, Chao Zhang","doi":"10.1137/23m1617400","DOIUrl":"https://doi.org/10.1137/23m1617400","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1078-1104, June 2025. <br/> Abstract. This paper addresses the challenge of constructing finite element [math] complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element [math] complexes. The spaces constructed are applied to discretize the quad curl problem, demonstrating optimal order of convergence. Furthermore, a hybridization technique is proposed, demonstrating its equivalence to nonconforming finite elements and weak Galerkin methods.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"125 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143979530","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}

{"title":"Spectral ACMS: A Robust Localized Approximated Component Mode Synthesis Method","authors":"Alexandre L. Madureira, Marcus Sarkis","doi":"10.1137/24m1665362","DOIUrl":"https://doi.org/10.1137/24m1665362","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1055-1077, June 2025. <br/> Abstract. We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous [math] coefficients. The methods are of Galerkin type and follow the Variational Multiscale and Localized Orthogonal Decomposition (LOD) approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, but it is not robust with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143933581","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}

{"title":"Density Estimation for Elliptic PDE with Random Input by Preintegration and Quasi-Monte Carlo Methods","authors":"Alexander D. Gilbert, Frances Y. Kuo, Abirami Srikumar","doi":"10.1137/24m1640070","DOIUrl":"https://doi.org/10.1137/24m1640070","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1025-1054, June 2025. <br/> Abstract. In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an [math]-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order [math] for arbitrarily small [math]. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143916087","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}

{"title":"A New Class of Splitting Methods That Preserve Ergodicity and Exponential Integrability for the Stochastic Langevin Equation","authors":"Chuchu Chen, Tonghe Dang, Jialin Hong, Fengshan Zhang","doi":"10.1137/24m1687686","DOIUrl":"https://doi.org/10.1137/24m1687686","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 1000-1024, April 2025. <br/> Abstract. In this paper, we propose a new class of splitting methods to solve the stochastic Langevin equation, which can simultaneously preserve the ergodicity and exponential integrability of the original equation. The central idea is to extract a stochastic subsystem that possesses the strict dissipation from the original equation, which is inspired by the inheritance of the Lyapunov structure for obtaining the ergodicity. We prove that the exponential moment of the numerical solution is bounded, thus validating the exponential integrability of the proposed methods. Further, we show that under moderate verifiable conditions, the methods have the first-order convergence in both strong and weak senses, and we present several concrete splitting schemes based on the methods. The splitting strategy of methods can be readily extended to construct conformal symplectic methods and high-order methods that preserve both the ergodicity and the exponential integrability, as demonstrated in numerical experiments. Our numerical experiments also show that the proposed methods have good performance in the long-time simulation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143884404","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}

{"title":"Reduced Krylov Basis Methods for Parametric Partial Differential Equations","authors":"Yuwen Li, Ludmil T. Zikatanov, Cheng Zuo","doi":"10.1137/24m1661236","DOIUrl":"https://doi.org/10.1137/24m1661236","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 976-999, April 2025. <br/> Abstract. This work is on a user-friendly reduced basis method for the solution of families of parametric partial differential equations by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and biconjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. We prove convergence estimates for the proposed method when the differential operator depends on two parameter coefficients and the preconditioner is the inverse of the operator at a fixed parameter. As is shown in numerical experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale parametrized systems in the reduced basis subspace. This reduces the computational cost by orders of magnitude, because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of problems restricted to the reduced basis subspace have much smaller sizes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"44 1","pages":"976-999"},"PeriodicalIF":2.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143876070","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}