In this manuscript, two-level methods applied to a symmetric interior penalty discontinuous Galerkin finite element discretization of a singularly perturbed reaction-diffusion equation are analyzed. Previous analyses of such methods have been performed numerically by Hemker et al. for the Poisson problem. The main innovation in this work is that explicit formulas for the optimal relaxation parameter of the two-level method for the Poisson problem in 1D are obtained, as well as very accurate closed form approximation formulas for the optimal choice in the reaction-diffusion case in all regimes. Using Local Fourier Analysis, performed at the matrix level to make it more accessible to the linear algebra community, it is shown that for DG penalization parameter values used in practice, it is better to use cell block-Jacobi smoothers of Schwarz type, in contrast to earlier results suggesting that point block-Jacobi smoothers are preferable, based on a smoothing analysis alone. The analysis also reveals how the performance of the iterative solver depends on the DG penalization parameter, and what value should be chosen to get the fastest iterative solver, providing a new, direct link between DG discretization and iterative solver performance. Numerical experiments and comparisons show the applicability of the expressions obtained in higher dimensions and more general geometries.
{"title":"Optimization of two-level methods for DG discretizations of reaction-diffusion equations","authors":"M. Gander, José Pablo Lucero Lorca","doi":"10.1051/m2an/2024059","DOIUrl":"https://doi.org/10.1051/m2an/2024059","url":null,"abstract":"In this manuscript, two-level methods applied to a symmetric\u0000 interior penalty discontinuous Galerkin finite element discretization\u0000 of a singularly perturbed reaction-diffusion equation are analyzed.\u0000 Previous analyses of such methods have been performed numerically by\u0000 Hemker et al. for the Poisson problem.\u0000 The main innovation in this work is that explicit formulas for the\u0000 optimal relaxation parameter of the two-level method for the Poisson\u0000 problem in 1D are obtained, as well as very accurate closed form\u0000 approximation formulas for the optimal choice in the\u0000 reaction-diffusion case in all regimes.\u0000 Using Local Fourier Analysis, performed at the matrix level to make\u0000 it more accessible to the linear algebra community, it is shown that\u0000 for DG penalization parameter values used in practice, it is better to\u0000 use cell block-Jacobi smoothers of Schwarz type, in contrast to\u0000 earlier results suggesting that point block-Jacobi smoothers\u0000 are preferable, based on a smoothing analysis alone.\u0000 The analysis also reveals how the performance of the iterative\u0000 solver depends on the DG penalization parameter, and what value should\u0000 be chosen to get the fastest iterative solver, providing a new, direct\u0000 link between DG discretization and iterative solver performance.\u0000 Numerical experiments and comparisons show the applicability of the\u0000 expressions obtained in higher dimensions and more general geometries.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141796345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We approximate a nonlinear multidimensional conservation law by Lattice Boltzmann Methods (LBM), based on underlying BGK type systems with finite number of velocities discretized by a transport-collision scheme. The collision part involves a relaxation parameter ω which value greatly influences the stability and accuracy of the method, as noted by many authors. In this article we clarify the relationship between ω and the parameters of the kinetic model and we highlight some new monotonicity properties which allow us to extend the previously obtained stability and convergence results. Numerical experiments are performed.
{"title":"Convergence of lattice Boltzmann methods with overrelaxation\u0000\u0000 for a nonlinear conservation law","authors":"Denise Aregba-Driollet","doi":"10.1051/m2an/2024058","DOIUrl":"https://doi.org/10.1051/m2an/2024058","url":null,"abstract":"We approximate a nonlinear multidimensional conservation law by Lattice Boltzmann Methods (LBM), based on underlying BGK type systems with finite number of velocities discretized by a transport-collision scheme. The collision part involves a relaxation parameter ω which value greatly influences the stability and accuracy of the method, as noted by many authors. In this article we clarify the relationship between ω and the parameters of the kinetic model and we highlight some new monotonicity properties which allow us to extend the previously obtained stability and convergence results. Numerical experiments are performed.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141650455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Patrick, Jr. Ciarlet, M. Kachanovska, Étienne Peillon
In this manuscript, we study solutions to resonant Maxwell’s equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω2u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [12] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series of numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.
在本手稿中，我们研究了异质等离子体中共振麦克斯韦方程的解。我们专注于上混合加热现象，这种现象发生在电磁波向粒子传递能量的局部区域。在二维情况下，它可以用偏微分方程 - div (α∇u) - ω2u = 0 进行数学建模，其中系数 α 是一个平滑的、符号变化的实值函数。由于符号变化的位置位于等离子体内部，因此方程是非椭圆的，而且是退化的。另一方面，利用极限吸收原理，我们可以建立一个近似于退化方程的椭圆方程组。那么，一个自然的问题是，如果存在退化方程的解，如何将其与椭圆方程的解族联系起来。为此，我们假定解的族收敛到一个极限，这个极限可分为正则部分和奇异部分，而这个极限吸收解受上文介绍的非椭圆方程支配。困难之一在于如何定义适当的规范和函数空间，以便研究非椭圆方程及其解。作为起点，我们重温了 A. Nicolopoulos、M. Campos Pinto、B. Després 和 P. Ciarlet Jr.之前关于这一主题的研究[12]，他们提出了等离子体加热问题的变分公式。我们改进了他们获得的结果，特别是通过对函数空间的不同选择，建立了解的存在性和唯一性。此外，我们还提出了一系列数值测试，将尼科洛普洛斯等人的数值结果与我们的数值方法得出的结果进行比较，我们发现后者的收敛性更好。
{"title":"Study of a degenerate non-elliptic equation to model plasma heating","authors":"Patrick, Jr. Ciarlet, M. Kachanovska, Étienne Peillon","doi":"10.1051/m2an/2024053","DOIUrl":"https://doi.org/10.1051/m2an/2024053","url":null,"abstract":"In this manuscript, we study solutions to resonant Maxwell’s equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω2u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [12] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series of numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141653414","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work focuses on the mixed formulation of linear wave equations. It provides a proof of stability and convergence of time discretisation of a semi discrete linear wave equation in mixed form with Störmer-Verlet time integration, that is uniform as the time step reaches its largest allowed value for stability (Courant-Friedrich-Levy condition), contrary to the proofs recalled here from the literature.
{"title":"Stability and space/time convergence of Störmer-Verlet time integration of the mixed formulation of linear wave equations","authors":"J. Chabassier","doi":"10.1051/m2an/2024047","DOIUrl":"https://doi.org/10.1051/m2an/2024047","url":null,"abstract":"This work focuses on the mixed formulation of linear wave equations. It provides a proof of stability and convergence of time discretisation of a semi discrete linear wave equation in mixed form with Störmer-Verlet time integration, that is uniform as the time step reaches its largest allowed value for stability (Courant-Friedrich-Levy condition), contrary to the proofs recalled here from the literature.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141338722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k−1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator. 1991 Mathematics Subject Classification 65N12, 65N30, 65N50, 76N05.
本研究的目的是针对具有奇异热源的广义布森斯克方程提出并分析一种混合非连续伽勒金（HDG）方法。我们使用 k、k-1 和 k 阶多项式来逼近速度、压力和温度。通过为压力引入拉格朗日乘法器，HDG 方法得到的近似速度场被证明是完全无发散和符合 H(div)的。在问题数据和解的小性假设下，我们通过布劳威尔定点定理证明了离散系统在两个维度上有一个解。如果进一步假设粘度和热导率为正常数，则可得出收敛率为 O(h)的先验误差估计和高效可靠的后验误差估计。最后用数值示例说明了理论分析，并展示了所获得的后验误差估计的性能。1991 年数学学科分类65N12，65N30，65N50，76N05。
{"title":"An exactly divergence-free hybridized discontinuous Galerkin method for the generalized Boussinesq equations with singular heat source","authors":"Haitao Leng","doi":"10.1051/m2an/2024037","DOIUrl":"https://doi.org/10.1051/m2an/2024037","url":null,"abstract":"The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k−1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer’s fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator.\u0000\u00001991 Mathematics Subject Classification\u0000\u000065N12, 65N30, 65N50, 76N05.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140961841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks.
{"title":"An explicit well-balanced scheme on staggered grids for barotropic Euler equations","authors":"Thierry Goudon, Sebastian Minjeaud","doi":"10.1051/m2an/2024035","DOIUrl":"https://doi.org/10.1051/m2an/2024035","url":null,"abstract":"In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140961843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with optimal design problems in the setting of the KirchhoffLove model for pure bending of a thin solid symmetric plate under a transverse load. For two isotropic elastic materials with a prescribed amount, the goal is to find their rearrangement within the domain that forms a least compliant structure. The homogenization method is used as a relaxation tool to overcome the lack of a classical solution of optimal design problem. Neccessary conditions of optimality were derived and an optimality criteria method for the single state compliance minimization problems is developed and tested on several examples.
{"title":"Optimal design of elastic plates","authors":"Ivana Crnjac, Jelena Jankov, Petar Kunštek","doi":"10.1051/m2an/2024036","DOIUrl":"https://doi.org/10.1051/m2an/2024036","url":null,"abstract":"This paper is concerned with optimal design problems in the setting of the KirchhoffLove model for pure bending of a thin solid symmetric plate under a transverse load. For two isotropic elastic materials with a prescribed amount, the goal is to find their rearrangement within the domain that forms a least compliant structure. The homogenization method is used as a relaxation tool to overcome the lack of a classical solution of optimal design problem. Neccessary conditions of optimality were derived and an optimality criteria method for the single state compliance minimization problems is developed and tested on several examples.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140971429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the finite element analysis for the Bean-Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with $L^1$-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D $H(curl)$-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.
本文致力于对受三维麦克斯韦方程支配的 Bean-Kim 模型进行有限元分析。该模型在宏观层面上描述了 II 型超导性，它导致了一个具有挑战性的耦合系统，该系统由法拉第方程和具有 $L^1$ 型非线性的第二类双曲准变不等式 (QVI) 组成，该非线性明确来自临界电流中的磁场依赖性。由于在 3D $H(curl)$ 设置中涉及麦克斯韦耦合，双曲 QVI 特性成为数值研究的主要挑战。本文提出了两种基于隐式欧拉和跃迁时间步法的混合有限元方法。一方面，隐式欧拉法产生了一个具有一阶卷曲型非线性的非标准卷曲-卷曲椭圆 QVI 系统。虽然该系统的良好拟合得到了保证，但其数值实现并不简单，需要使用计算复杂度较高的两阶段迭代过程。另一方面，通过在两个不同的时间步长上逼近电场和磁场，跃迁法自然消除了臭名昭著的 QVI 结构，因而更为合适。更重要的是，利用合适的次微分和优化技术，我们能够证明其精确解在电场方面的高效可计算的显式公式，这使得其数值计算比欧拉法更为有利。作为进一步的优势，跃迁法适用于涉及应用电流源和温度分布在时间上有界变化（BV）的低规则数据的广泛场景。通过针对 BV 数据的非标准技术论证，我们的分析证明了所提出的蛙跳法的条件稳定性以及最终的均匀收敛性。本文最后通过三维数值测试展示了所提数值解决方案的合理性和高效性。
{"title":"Numerical solutions to hyperbolic Maxwell quasi-variational inequalities in Bean-Kim model for type-II superconductivity","authors":"Maurice Hensel, Malte Winckler, Irwin Yousept","doi":"10.1051/m2an/2024034","DOIUrl":"https://doi.org/10.1051/m2an/2024034","url":null,"abstract":"This paper is devoted to the finite element analysis for the Bean-Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with $L^1$-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D $H(curl)$-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140966636","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell’s transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues.
{"title":"Error estimates for a mixed finite element method for the Maxwell’s transmission eigenvalue problem","authors":"Chao Wang, Jintao Cui, Jiguang Sun","doi":"10.1051/m2an/2024033","DOIUrl":"https://doi.org/10.1051/m2an/2024033","url":null,"abstract":"In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell’s transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues.","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141008709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Charles-Édouard Bréhier, David Cohen, Johan Ulander
We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity. p, li { white-space: pre-wrap; }
{"title":"Analysis of a positivity-preserving splitting scheme for some semilinear stochastic heat equations","authors":"Charles-Édouard Bréhier, David Cohen, Johan Ulander","doi":"10.1051/m2an/2024032","DOIUrl":"https://doi.org/10.1051/m2an/2024032","url":null,"abstract":"We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of semilinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.\u0000\u0000\u0000p, li { white-space: pre-wrap; }","PeriodicalId":505020,"journal":{"name":"ESAIM: Mathematical Modelling and Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141017040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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