{"title":"Error estimates for a class of continuous Bonse-type inequalities","authors":"D. Marques, P. Trojovský","doi":"10.1090/mcom/3741","DOIUrl":"https://doi.org/10.1090/mcom/3741","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76407424","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":"Algorithms for fundamental invariants and equivariants of finite groups","authors":"E. Hubert, Erick D. Rodríguez Bazan","doi":"10.1090/mcom/3749","DOIUrl":"https://doi.org/10.1090/mcom/3749","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73634498","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":"Finite element/holomorphic operator function method for the transmission eigenvalue problem","authors":"Bo Gong, Jiguang Sun, T. Turner, Chunxiong Zheng","doi":"10.1090/mcom/3767","DOIUrl":"https://doi.org/10.1090/mcom/3767","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73094539","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 Galerkin finite element solution uh of the Possion equation −∆u = f under the Neumann boundary condition in a possibly nonconvex polygon Ω, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: ‖uh‖L∞(Ω) ≤ C`h‖u‖L∞(Ω), where `h = ln(2+1/h) for piecewise linear elements and `h = 1 for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds: ‖u− uh‖L∞(Ω) ≤ C`h‖u− Ihu‖L∞(Ω), where Ih denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm: ‖u− uh‖L∞(Ω) ≤ { C`hh k+2− 2 p ‖f‖Wk,p(Ω) if r ≥ k + 1, C`hh ‖f‖Hk(Ω) if r = k, where r ≥ 1 is the degree of finite elements, k is any nonnegative integer no larger than r, and p ∈ [2,∞) can be arbitrarily large.
{"title":"Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh","authors":"Buyang Li","doi":"10.1090/mcom/3724","DOIUrl":"https://doi.org/10.1090/mcom/3724","url":null,"abstract":"The Galerkin finite element solution uh of the Possion equation −∆u = f under the Neumann boundary condition in a possibly nonconvex polygon Ω, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: ‖uh‖L∞(Ω) ≤ C`h‖u‖L∞(Ω), where `h = ln(2+1/h) for piecewise linear elements and `h = 1 for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds: ‖u− uh‖L∞(Ω) ≤ C`h‖u− Ihu‖L∞(Ω), where Ih denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm: ‖u− uh‖L∞(Ω) ≤ { C`hh k+2− 2 p ‖f‖Wk,p(Ω) if r ≥ k + 1, C`hh ‖f‖Hk(Ω) if r = k, where r ≥ 1 is the degree of finite elements, k is any nonnegative integer no larger than r, and p ∈ [2,∞) can be arbitrarily large.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82742442","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 consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability estimates for the evolution equation. A posteriori error estimates in $L^2(H)$ and $L^infty (H)$ norms of optimal order are derived. Extensive numerical experiments indicate the reliability and the optimality of the estimators for the schemes considered, as well as their efficiency as error indicators driving adaptive mesh selection locating singularities of the problem.
{"title":"A posteriori error analysis for approximations of time-fractional subdiffusion problems","authors":"L. Banjai, C. Makridakis","doi":"10.1090/mcom/3723","DOIUrl":"https://doi.org/10.1090/mcom/3723","url":null,"abstract":"In this paper we consider a sub-diffusion problem where the fractional time derivative is approximated either by the L1 scheme or by Convolution Quadrature. We propose new interpretations of the numerical schemes which lead to a posteriori error estimates. Our approach is based on appropriate pointwise representations of the numerical schemes as perturbed evolution equations and on stability estimates for the evolution equation. A posteriori error estimates in $L^2(H)$ and $L^infty (H)$ norms of optimal order are derived. Extensive numerical experiments indicate the reliability and the optimality of the estimators for the schemes considered, as well as their efficiency as error indicators driving adaptive mesh selection locating singularities of the problem.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81776078","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 presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity of the proposed entropy, the Roe-type dissipation operator in terms of the entropy variables can be formulated. Furthermore, we construct two sets of second-order entropy preserving (EP) numerical fluxes for the reactive Euler equations. Based on the EP fluxes and the Roe-type dissipation operators, high-order EP/ES fluxes are derived. Numerical experiments validate the designed accuracy and good performance of our schemes for smooth and discontinuous initial value problems. The entropy decrease of ES schemes is verified as well.
{"title":"Strictly convex entropy and entropy stable schemes for reactive Euler equations","authors":"Weifeng Zhao","doi":"10.1090/mcom/3721","DOIUrl":"https://doi.org/10.1090/mcom/3721","url":null,"abstract":"This paper presents entropy analysis and entropy stable (ES) finite difference schemes for the reactive Euler equations with chemical reactions. For such equations we point out that the thermodynamic entropy is no longer strictly convex. To address this issue, we propose a strictly convex entropy function by adding an extra term to the thermodynamic entropy. Thanks to the strict convexity of the proposed entropy, the Roe-type dissipation operator in terms of the entropy variables can be formulated. Furthermore, we construct two sets of second-order entropy preserving (EP) numerical fluxes for the reactive Euler equations. Based on the EP fluxes and the Roe-type dissipation operators, high-order EP/ES fluxes are derived. Numerical experiments validate the designed accuracy and good performance of our schemes for smooth and discontinuous initial value problems. The entropy decrease of ES schemes is verified as well.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91232547","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 the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford relatively large time stepping and speed up the simulation. In practical numerical implementations it has been long observed that the resulting system exhibits remarkable stability properties in the regime where the stabilization parameter is O(1), the dissipation coefficient is vanishingly small and the size of the time step is moderately large. In this work we develop a new stability theory to address this perplexing phenomenon.
{"title":"Why large time-stepping methods for the Cahn-Hilliard equation is stable","authors":"Dong Li","doi":"10.1090/mcom/3768","DOIUrl":"https://doi.org/10.1090/mcom/3768","url":null,"abstract":"We consider the Cahn-Hilliard equation with standard double-well potential. We employ a prototypical class of first order in time semi-implicit methods with implicit treatment of the linear dissipation term and explicit extrapolation of the nonlinear term. When the dissipation coefficient is held small, a conventional wisdom is to add a judiciously chosen stabilization term in order to afford relatively large time stepping and speed up the simulation. In practical numerical implementations it has been long observed that the resulting system exhibits remarkable stability properties in the regime where the stabilization parameter is O(1), the dissipation coefficient is vanishingly small and the size of the time step is moderately large. In this work we develop a new stability theory to address this perplexing phenomenon.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82291311","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":"Stochastic gradient descent for linear inverse problems in Hilbert spaces","authors":"Shuai Lu, P. Mathé","doi":"10.1090/mcom/3714","DOIUrl":"https://doi.org/10.1090/mcom/3714","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81897724","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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