SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1369-1397, August 2025. Abstract. In this work, we investigate the Tikhonov-type regularized solutions and their finite element solutions to the inverse space-dependent source problem from boundary measurement data. First, with the classical source condition, we establish the convergence of regularized solutions and their finite element solutions under the standard [math] norm. The error estimates present explicit dependence on the critical parameters like noise level, regularization parameter, mesh size, and time step size. Next, based on a proposed weak norm, we get the stability of Lipschitz type for the inverse problem, and then the first order convergence of regularized solutions can be derived in the sense of weak norm. We get this convergence without any source condition. Moreover, this work is carried out for the discrete data. We suppose that the observation points are discrete and the pointwise measurement data come with independent sub-Gaussian random noises. Then we give the stochastic convergence of regularized solutions and propose an efficient iterative algorithm to determine the optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithms.
{"title":"Convergence Estimates of Regularized Solutions to Inverse Space-Dependent Source Problems with Time-Dependent Boundary Measurement","authors":"Chunlong Sun, Wenlong Zhang","doi":"10.1137/24m1692885","DOIUrl":"https://doi.org/10.1137/24m1692885","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1369-1397, August 2025. <br/> Abstract. In this work, we investigate the Tikhonov-type regularized solutions and their finite element solutions to the inverse space-dependent source problem from boundary measurement data. First, with the classical source condition, we establish the convergence of regularized solutions and their finite element solutions under the standard [math] norm. The error estimates present explicit dependence on the critical parameters like noise level, regularization parameter, mesh size, and time step size. Next, based on a proposed weak norm, we get the stability of Lipschitz type for the inverse problem, and then the first order convergence of regularized solutions can be derived in the sense of weak norm. We get this convergence without any source condition. Moreover, this work is carried out for the discrete data. We suppose that the observation points are discrete and the pointwise measurement data come with independent sub-Gaussian random noises. Then we give the stochastic convergence of regularized solutions and propose an efficient iterative algorithm to determine the optimal regularization parameter. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithms.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"8 1","pages":"1369-1397"},"PeriodicalIF":2.9,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144566011","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 63, Issue 3, Page 1344-1368, June 2025. Abstract. In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial and test spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the [math]-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of [math]-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound or, in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.
{"title":"Ultra-Weak Least Squares Discretizations for Unique Continuation and Cauchy Problems","authors":"Harald Monsuur, Rob Stevenson","doi":"10.1137/24m1674844","DOIUrl":"https://doi.org/10.1137/24m1674844","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1344-1368, June 2025. <br/> Abstract. In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial and test spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the [math]-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of [math]-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound or, in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144341262","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}
Mária Lukáčová-Medviďová, Bangwei She, Yuhuan Yuan
SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1254-1280, June 2025. Abstract. In the present paper, we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier–Stokes–Fourier system. The Monte Carlo method, frequently used for approximating statistical moments, is combined with a suitable deterministic discretization method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to the statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.
{"title":"Convergence Analysis of the Monte Carlo Method for the Random Navier–Stokes–Fourier System","authors":"Mária Lukáčová-Medviďová, Bangwei She, Yuhuan Yuan","doi":"10.1137/23m1563232","DOIUrl":"https://doi.org/10.1137/23m1563232","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1254-1280, June 2025. <br/> Abstract. In the present paper, we consider the initial data, external force, viscosity coefficients, and heat conductivity coefficient as random data for the compressible Navier–Stokes–Fourier system. The Monte Carlo method, frequently used for approximating statistical moments, is combined with a suitable deterministic discretization method in physical space and time. Under the assumption that numerical densities and temperatures are bounded in probability, we prove the convergence of random finite volume solutions to the statistical strong solution by applying genuine stochastic compactness arguments. Further, we show the convergence and error estimates for Monte Carlo estimators of the expectation and deviation. We present several numerical results to illustrate the theoretical results.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"38 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144311938","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 63, Issue 3, Page 1315-1343, June 2025. Abstract. We propose and rigorously analyze a finite element method for the approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments.
{"title":"Finite Element Approximation of Stationary Fokker–Planck–Kolmogorov Equations with Application to Periodic Numerical Homogenization","authors":"Timo Sprekeler, Endre Süli, Zhiwen Zhang","doi":"10.1137/24m1692848","DOIUrl":"https://doi.org/10.1137/24m1692848","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1315-1343, June 2025. <br/> Abstract. We propose and rigorously analyze a finite element method for the approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144311937","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}
Fabrice Deluzet, Clément Guillet, Jacek Narski, Paul Pace
SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1281-1314, June 2025. Abstract. Particle-in-cell (PIC) methods embedding sparse grids have been recently introduced to decrease the statistical noise inherent to PIC approximations. In sparse-PIC methods, the numerical noise is filtered out from the approximation thanks to a reconstruction of the grid quantities on a hierarchy of coarse meshes. This procedure introduces a significant gain in the precision of the numerical approximation with respect to the mean number of particles in a grid cell, this parameter controlling the numerical noise, but also introduces a slight discrepancy of the method precision with respect to the mesh resolution. In previous studies, this issue is addressed by a careful tuning of the grids composing the sparse grid hierarchy, to define a trade-off between the gain in the numerical noise and the loss in the grid error. The present work is dedicated to improving the precision of sparse-PIC methods with respect to the mesh resolution and, contrary to the previous achievements, without deteriorating the gains with respect to the statistical noise. A refined error estimate is proposed. It permits one to control the number of numerical particles to obtain a comparable statistical noise in the approximation carried out by either a standard or a sparse-PIC method (and thus assess the true merits of the methods).
{"title":"High-Order Sparse-PIC Methods: Analysis and Numerical Investigations","authors":"Fabrice Deluzet, Clément Guillet, Jacek Narski, Paul Pace","doi":"10.1137/24m1665143","DOIUrl":"https://doi.org/10.1137/24m1665143","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1281-1314, June 2025. <br/> Abstract. Particle-in-cell (PIC) methods embedding sparse grids have been recently introduced to decrease the statistical noise inherent to PIC approximations. In sparse-PIC methods, the numerical noise is filtered out from the approximation thanks to a reconstruction of the grid quantities on a hierarchy of coarse meshes. This procedure introduces a significant gain in the precision of the numerical approximation with respect to the mean number of particles in a grid cell, this parameter controlling the numerical noise, but also introduces a slight discrepancy of the method precision with respect to the mesh resolution. In previous studies, this issue is addressed by a careful tuning of the grids composing the sparse grid hierarchy, to define a trade-off between the gain in the numerical noise and the loss in the grid error. The present work is dedicated to improving the precision of sparse-PIC methods with respect to the mesh resolution and, contrary to the previous achievements, without deteriorating the gains with respect to the statistical noise. A refined error estimate is proposed. It permits one to control the number of numerical particles to obtain a comparable statistical noise in the approximation carried out by either a standard or a sparse-PIC method (and thus assess the true merits of the methods).","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"600 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144319836","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 63, Issue 3, Page 1232-1253, June 2025. Abstract. This work investigates the convergence of a domain decomposition method for the Poisson–Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the [math] norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson–Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [C. Quan, B. Stamm, and Y. Maday, SIAM J. Sci. Comput., 41 (2019), pp. B320–B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.
SIAM数值分析杂志，第63卷，第3期，1232-1253页，2025年6月。摘要。这项工作研究了泊松-玻尔兹曼模型的区域分解方法的收敛性，该模型可以表述为内部-外部传输问题。为了研究其收敛性，我们引入了一个内外常数，给出了内调和函数的[数学]范数的上界，并建立了相关Dirichlet-to-Neumann算子的谱等价来估计内外迭代算子的谱。由于外子域的无界性，这种分析是不平凡的，这与经典的具有非重叠有界子域的Schwarz交替方法分析不同。在现实中证明了线性泊松-玻尔兹曼溶剂模型，当松弛参数在0 ~ 2之间时，保证了内外迭代的收敛性。这一收敛结果解释了[C]中开发的ddLPB方法的良好性能。Quan， B. Stamm和Y. Maday, SIAM J. Sci。第一版。[j], 41 (2019), pp. B320-B350]，其中松弛参数设置为1。数值模拟验证了我们的收敛性分析，并研究了内外迭代的最优松弛参数。
{"title":"Convergence Analysis of a Solver for the Linear Poisson–Boltzmann Model","authors":"Xuanyu Liu, Yvon Maday, Chaoyu Quan, Hui Zhang","doi":"10.1137/24m1717087","DOIUrl":"https://doi.org/10.1137/24m1717087","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1232-1253, June 2025. <br/> Abstract. This work investigates the convergence of a domain decomposition method for the Poisson–Boltzmann model that can be formulated as an interior-exterior transmission problem. To study its convergence, we introduce an interior-exterior constant providing an upper bound of the [math] norm of any harmonic function in the interior, and establish a spectral equivalence for related Dirichlet-to-Neumann operators to estimate the spectrum of interior-exterior iteration operator. This analysis is nontrivial due to the unboundedness of the exterior subdomain, which distinguishes it from the classical analysis of the Schwarz alternating method with nonoverlapping bounded subdomains. It is proved that for the linear Poisson–Boltzmann solvent model in reality, the convergence of interior-exterior iteration is ensured when the relaxation parameter lies between 0 and 2. This convergence result interprets the good performance of ddLPB method developed in [C. Quan, B. Stamm, and Y. Maday, SIAM J. Sci. Comput., 41 (2019), pp. B320–B350] where the relaxation parameter is set to 1. Numerical simulations are conducted to verify our convergence analysis and to investigate the optimal relaxation parameter for the interior-exterior iteration.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"140 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144260242","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 63, Issue 3, Page 1209-1231, June 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1183-1208, June 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1160-1182, June 2025. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 3, Page 1128-1159, June 2025. 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.
{"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}
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