ESAIM: Mathematical Modelling and Numerical Analysis最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}