W. Dörfler, S. Findeisen, C. Wieners, Daniel Ziegler
{"title":"2. Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves","authors":"W. Dörfler, S. Findeisen, C. Wieners, Daniel Ziegler","doi":"10.1515/9783110548488-002","DOIUrl":"https://doi.org/10.1515/9783110548488-002","url":null,"abstract":"","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130658492","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":"1. Space-time boundary element methods for the heat equation","authors":"S. Dohr, O. Steinbach, K. Niino","doi":"10.1515/9783110548488-001","DOIUrl":"https://doi.org/10.1515/9783110548488-001","url":null,"abstract":"","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132210086","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":"4. A space-time DPG method for the wave equation in multiple dimensions","authors":"J. Gopalakrishnan, Paulina Sepúlveda","doi":"10.1515/9783110548488-004","DOIUrl":"https://doi.org/10.1515/9783110548488-004","url":null,"abstract":"","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124068441","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":"3. A space-time discontinuous Petrov–Galerkin method for acoustic waves","authors":"Johannes Ernesti, C. Wieners","doi":"10.1515/9783110548488-003","DOIUrl":"https://doi.org/10.1515/9783110548488-003","url":null,"abstract":"","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116921807","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":"7. Space-time finite element methods for parabolic evolution equations: discretization, a posteriori error estimation, adaptivity and solution","authors":"O. Steinbach, Huidong Yang","doi":"10.1515/9783110548488-007","DOIUrl":"https://doi.org/10.1515/9783110548488-007","url":null,"abstract":"In this work, we present an overview on the development of space–time finite element methods for the numerical solution of some parabolic evolution equations with the heat equation as a model problem. Instead of using more standard semi– discretization approaches such as the method of lines or Rothe’s method, our specific focus is on continuous space–time finite element discretizations in space and time simultaneously. While such discretizations bring more flexibility to the space–time finite element error analysis and error control, they usually lead to higher computational complexity and memory consumptions in comparison with standard time– stepping methods. Therefore, progress on a posteriori error estimation and respective adaptive schemes in the space–time domain is reviewed, which aims to save a number of degrees of freedom, and hence reduces complexity, and recovers optimal order error estimates. Further, we provide a summary on recent advances in efficient parallel space–time iterative solution strategies for the related large–scale linear systems of algebraic equations, that are crucial to make such all–at–once approaches competitive with traditional time stepping methods. Finally, some numerical results are given to demonstrate the benefits of a particular adaptive space–time finite element method, the robustness of some space–time algebraic multigrid methods, and the applicability of space–time finite element methods for the solution of some parabolic optimal control problem.","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117231856","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":"5. Adaptive space-time isogeometric analysis for parabolic evolution problems","authors":"U. Langer, S. Matculevich, S. Repin","doi":"10.1515/9783110548488-005","DOIUrl":"https://doi.org/10.1515/9783110548488-005","url":null,"abstract":"The paper is concerned with locally stabilized space-time IgA approximations to initial boundary value problems of the parabolic type. Originally, similar schemes (but weighted with a global mesh parameter) was presented and studied by U. Langer, M. Neumueller, and S. Moore (2016). The current work devises a localised version of this scheme and establishes coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with the corresponding approximation error estimates for B-splines, we show that the space-time IgA solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. The adaptive mesh refinement algorithm proposed in the paper is based on a posteriori error estimates of the functional type that has been rigorously studied in earlier works by S. Repin (2002) and U. Langer, S. Matculevich, and S. Repin (2017). Numerical results presented in the second part of the paper confirm the improved convergence of global approximation errors. Moreover, these results also confirm the local efficiency of the error indicators produced by the error majorants.","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"32 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120819566","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":"6. Generating admissible space-time meshes for moving domains in (d + 1) dimensions","authors":"Elias Karabelas, Martin Neumuller","doi":"10.1515/9783110548488-006","DOIUrl":"https://doi.org/10.1515/9783110548488-006","url":null,"abstract":"In this paper we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this we will present an algorithm to generate $d+1$-dimensional simplex space-time meshes and we show under natural assumptions that the resulting space-time meshes are admissible. Further we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented.","PeriodicalId":313981,"journal":{"name":"Space-Time Methods","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121614660","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}