On Helmholtz decompositions and their generalizations-An overview

Mathematical Methods in the Applied Sciences

Volumn 33, Issue 4, 2010, Pages 374-383

  Sprossig W ^a  

^a TU BERGAKADEMIE FREIBERG   (Germany)

Author keywords

Bergman projections; Helmholtz decomposition; Hodge decomposition; Quaternionic operator calculus

Indexed keywords

1-FORMS; CLIFFORD; HELMHOLTZ; HELMHOLTZ DECOMPOSITION; HILBERT; HODGE DECOMPOSITION; NUMERICAL SOLUTION; OPERATOR CALCULUS; PROJECTION METHOD; SOBOLEV SPACE;

MATHEMATICAL OPERATORS; NUMERICAL ANALYSIS; NUMERICAL METHODS; SOBOLEV SPACES; VECTOR SPACES;

DECOMPOSITION;

EID: 77249131799     PISSN: 01704214     EISSN: 10991476     Source Type: Journal    
DOI: 10.1002/mma.1212     Document Type: Article

References (25)

1. Blumenthal O. Über die Zerlegung unendlicher Vektorfelder. Mathematische Annalen 1905; 61(2):235-250.
2. Bolik J, von Wahl W. Estimating ∇u in terms of ÷u, curlu, either (ny·u) or ny×u and the Topology. Mathematical Methods in the Applied Sciences 1997; 20:737-744.
3. Denaro FM. On the application of the Helmholtz-Hodge decomposition in projection methods for the numerical solution of the incompressible Navier-Stokes equations with general boundary conditions. International Journal for Numerical Methods in Fluids 2003; 43:43-69.
4. Mitrea D, Mitrea M. Sharp Hodge decompositions in two and three dimensional Lipschitz domains. Comptes Rendus, Tom 334, fasciule 2, Analyse Mathematique 2002; 109-112. (Pubitemid 36645546)
5. Schwarz G. Hodge Decompositions-A Method for Solving Boundary Value Problems. LMN, vol. 1607. Springer: Berlin, 1995.
6. Hofmann S, Mitrea M, Taylor M. Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains and other classes of finite perimeter domains. The Journal of Geometric Analysis 2007; 17(4):593-647.
7. Picard R. On the boundary value problem of electro- and magneto-statics. Proceedings of the Royal Society of Edinburg 1982; 165-174.
8. Hamilton WR. Elements of Quaternions. Longmans Green: London, 1969. Reprinted by Chelsea, New York, 1866.
9. Blaschke W. Anwendung dualer Quaternionen auf Kinematik. Annales Academiæ Scientiarum Fennicae, Ser. A1, Mathematica 1958; 250(3).
10. Gürlebeck K, Sprössig W. Quaternionic Analysis and Boundary Value Problems. Birkhäuser: Basel, 1990.
11. Gürlebeck K, Sprössig W. Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley: Chichester, 1997.
12. Sprössig W. On decomposition of the Clifford Valued Hilbert Space and their Applications to Boundary Value Problems. AACA 1995; 5(2):167-185.
13. Sprössig W, Guerlebeck K. Representation theory for classes of initial value problems with quaternionic analysis. Mathematical Methods in the Applied Sciences 2002; 25:1371-1382.
14. Sprössig W. Fluid flow equations with variable viscosity in quaternionic setting. AACA 2007; 17:259-272.
15. Sprössig W. Fluid flow equations with thermal effects. ACAA 2009; accepted for publication (ISSN 0188-7009).
16. Sprössig W. Quaternionic analysis and Maxwell's equations. Cubo, A mathematical Journal 2005; 7(2):57-67.
17. Van Lancker P. Clifford analysis on the unit sphere. Thesis, University of Ghent, 1997.
18. Buffa A. Hodge decomposition on the boundary of a polyhedron: the multi-connected case. Mathematical Models and Methods in Applied Sciences 2001; 11(9):1491-1504.
19. Buffa A, Ciarlet Jr P. On traces for functional spaces related to Maxwell's equation. Part I: an integration by parts formula in Lipschitz polyhedra. Mathematical Methods in the Applied Sciences 2001; 24(1):9-30.
20. Buffa A, Ciarlet Jr P. n traces for functional spaces related to Maxwell's equation. Part II: Hodge decompositions on the boundary of Lipchitz polyhedra and applications. Mathematical Methods in the Applied Sciences 2001; 24(1):31-48.
21. Malonek HR, Ren G. Almansi-type theorems in Clifford analysis. Mathematical Methods in the Applied Sciences 2002; 25:1541-1552.
22. Aronszajn N, Creese TM, Lipkin L J. Polyharmonic Functions. Clarendon Press: Oxford, 1983.
23. Picard R. Some decomposition theorems and their application to non-linear potential theory and Hodge theory. Mathematical Methods in the Applied Sciences 1989; 12(1):35-52.
24. Amrouche C, Bernardi C, Dauge M, Girault V. Vector potentials in three-dimensional nonsmooth domains. Mathematical Methods in the Applied Sciences 1998; 21:823-864.
25. Sprössig W. Quaternionic operator methods in fluid dynamics. ACAA 2008; 18:963-978.