# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. (English) Zbl 1153.78006
Summary: We develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of $𝐇\left(\mathrm{𝐜𝐮𝐫𝐥},{\Omega }\right)$- and $𝐇\left(\mathrm{𝐝𝐢𝐯},{\Omega }\right)$-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 $𝐇\left(\mathrm{𝐜𝐮𝐫𝐥},{\Omega }\right)$ and $𝐇\left(\mathrm{𝐝𝐢𝐯},{\Omega }\right)$. Our preconditioner for $𝐇\left(\mathrm{𝐜𝐮𝐫𝐥},{\Omega }\right)$ 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)].

##### MSC:
 78M10 Finite element methods (optics) 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE) 65N55 Multigrid methods; domain decomposition (BVP of PDE) 65N22 Solution of discretized equations (BVP of PDE) 65F10 Iterative methods for linear systems