Pub Date : 2023-01-01Epub Date: 2022-11-16DOI: 10.1007/s00211-022-01334-8
Roland Becker, Gregor Gantner, Michael Innerberger, Dirk Praetorius
We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.
{"title":"Goal-oriented adaptive finite element methods with optimal computational complexity.","authors":"Roland Becker, Gregor Gantner, Michael Innerberger, Dirk Praetorius","doi":"10.1007/s00211-022-01334-8","DOIUrl":"10.1007/s00211-022-01334-8","url":null,"abstract":"<p><p>We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"153 1","pages":"111-140"},"PeriodicalIF":2.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9829645/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10536358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.1007/s00211-022-01342-8
Alexander Rieder
We introduce a novel discretization technique for both elliptic and parabolic fractional diffusion problems based on double exponential quadrature formulas and the Riesz-Dunford functional calculus. Compared to related schemes, the new method provides faster convergence with fewer parameters that need to be adjusted to the problem. The scheme takes advantage of any additional smoothness in the problem without requiring a-priori knowledge to tune parameters appropriately. We prove rigorous convergence results for both, the case of finite regularity data as well as for data in certain Gevrey-type classes. We confirm our findings with numerical tests.
{"title":"Double exponential quadrature for fractional diffusion.","authors":"Alexander Rieder","doi":"10.1007/s00211-022-01342-8","DOIUrl":"https://doi.org/10.1007/s00211-022-01342-8","url":null,"abstract":"<p><p>We introduce a novel discretization technique for both elliptic and parabolic fractional diffusion problems based on double exponential quadrature formulas and the Riesz-Dunford functional calculus. Compared to related schemes, the new method provides faster convergence with fewer parameters that need to be adjusted to the problem. The scheme takes advantage of any additional smoothness in the problem without requiring a-priori knowledge to tune parameters appropriately. We prove rigorous convergence results for both, the case of finite regularity data as well as for data in certain Gevrey-type classes. We confirm our findings with numerical tests.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"153 2-3","pages":"359-410"},"PeriodicalIF":2.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9998606/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9472166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-28DOI: 10.1007/s00211-022-01340-w
Jean-Baptiste Bellet, M. Brachet, J. Croisille
{"title":"Interpolation on the cubed sphere with spherical harmonics","authors":"Jean-Baptiste Bellet, M. Brachet, J. Croisille","doi":"10.1007/s00211-022-01340-w","DOIUrl":"https://doi.org/10.1007/s00211-022-01340-w","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"153 1","pages":"249 - 278"},"PeriodicalIF":2.1,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45406415","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}
Pub Date : 2022-12-26DOI: 10.1007/s00211-022-01338-4
C. Carstensen, M. Schedensack
{"title":"Two discretisations of the time-dependent Bingham problem","authors":"C. Carstensen, M. Schedensack","doi":"10.1007/s00211-022-01338-4","DOIUrl":"https://doi.org/10.1007/s00211-022-01338-4","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"153 1","pages":"411-450"},"PeriodicalIF":2.1,"publicationDate":"2022-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44745004","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}
Pub Date : 2022-12-24DOI: 10.1007/s00211-022-01336-6
Elisa Calzola, E. Carlini, Xavier Dupuis, Francisco J. Silva
{"title":"A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions","authors":"Elisa Calzola, E. Carlini, Xavier Dupuis, Francisco J. Silva","doi":"10.1007/s00211-022-01336-6","DOIUrl":"https://doi.org/10.1007/s00211-022-01336-6","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"153 1","pages":"49-84"},"PeriodicalIF":2.1,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42971572","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}
Pub Date : 2022-11-09DOI: 10.1007/s00211-022-01330-y
Chiara Fuda, R. Campagna, K. Hormann
{"title":"Publisher Correction to: On the numerical stability of linear barycentric rational interpolation","authors":"Chiara Fuda, R. Campagna, K. Hormann","doi":"10.1007/s00211-022-01330-y","DOIUrl":"https://doi.org/10.1007/s00211-022-01330-y","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"152 1","pages":"787 - 788"},"PeriodicalIF":2.1,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45996944","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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