{"title":"A new class of entropy stable schemes for hyperbolic systems: Finite element methods","authors":"I. Gkanis, C. Makridakis","doi":"10.1090/mcom/3617","DOIUrl":"https://doi.org/10.1090/mcom/3617","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89565212","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 approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative","authors":"J. Jorgenson, L. Smajlovic, H. Then","doi":"10.1090/mcom/3619","DOIUrl":"https://doi.org/10.1090/mcom/3619","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88525115","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":"Primes that become composite after changing an arbitrary digit","authors":"M. Filaseta, Jeremiah T. Southwick","doi":"10.1090/mcom/3593","DOIUrl":"https://doi.org/10.1090/mcom/3593","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84932213","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":"On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations","authors":"V. John, P. Knobloch, Paul Korsmeier","doi":"10.1090/mcom/3576","DOIUrl":"https://doi.org/10.1090/mcom/3576","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77204870","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 show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary states in contrast with splitting methods in general. We prove these results when the source term only depends on the space variable. Numerical experiments suggest that the second order convergence persists with general nonlinearities.
{"title":"Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with oblique boundary conditions","authors":"Guillaume Bertoli, C. Besse, G. Vilmart","doi":"10.1090/MCOM/3664","DOIUrl":"https://doi.org/10.1090/MCOM/3664","url":null,"abstract":"We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary states in contrast with splitting methods in general. We prove these results when the source term only depends on the space variable. Numerical experiments suggest that the second order convergence persists with general nonlinearities.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88032098","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 propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.
{"title":"Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations","authors":"C. Cancès, Flore Nabet, M. Vohralík","doi":"10.1090/MCOM/3577","DOIUrl":"https://doi.org/10.1090/MCOM/3577","url":null,"abstract":"We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82092532","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 study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.
{"title":"Analysis of finite element methods for surface vector-Laplace eigenproblems","authors":"A. Reusken","doi":"10.1090/mcom/3728","DOIUrl":"https://doi.org/10.1090/mcom/3728","url":null,"abstract":"In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78286539","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 consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.
{"title":"Abel maps for nodal curves via tropical geometry","authors":"Alex Abreu, Sally Andria, M. Pacini","doi":"10.1090/mcom/3717","DOIUrl":"https://doi.org/10.1090/mcom/3717","url":null,"abstract":"We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77117567","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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