%0 Journal Article %1 hiptmair2007nodal %A Hiptmair, Ralf %A Xu, Jinchao %D 2007 %J SIAM Journal on Numerical Analysis %K 65f10-iterative-methods-for-linear-systems 65n22-pdes-bvps-solution-of-discretized-equations 65n30-pdes-bvps-finite-elements 65n55-pdes-bvps-multigrid-methods-domain-decomposition %N 6 %R 10.1137/060660588 %T Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces %U https://epubs.siam.org/doi/10.1137/060660588 %V 45 %X In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div, )-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, ) and H(div, ). Our preconditioner for H(curl, ) is similar to an algorithm proposed in R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999.

@article{hiptmair2007nodal, abstract = {In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, )- and H(div, )-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, ) and H(div, ). Our preconditioner for H(curl, ) is similar to an algorithm proposed in [R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999].}, added-at = {2023-11-09T03:45:16.000+0100}, author = {Hiptmair, Ralf and Xu, Jinchao}, biburl = {https://www.bibsonomy.org/bibtex/258871f49cf3f648445e87d969820e6c1/gdmcbain}, doi = {10.1137/060660588}, interhash = {0b01ae862b4224d4cac4c559e4d193ea}, intrahash = {58871f49cf3f648445e87d969820e6c1}, journal = {SIAM Journal on Numerical Analysis}, keywords = {65f10-iterative-methods-for-linear-systems 65n22-pdes-bvps-solution-of-discretized-equations 65n30-pdes-bvps-finite-elements 65n55-pdes-bvps-multigrid-methods-domain-decomposition}, number = 6, timestamp = {2023-11-09T03:45:16.000+0100}, title = {Nodal Auxiliary Space Preconditioning in {H}(curl) and {H}(div) Spaces }, url = {https://epubs.siam.org/doi/10.1137/060660588}, volume = 45, year = 2007 }