×

Discontinuous finite element basis functions for nonlinear partial differential equations. (English) Zbl 0702.65083

The article is claimed by its authors to introduce for the first time discontinuous finite element basis functions in a rigorous way for the approximative solution of nonlinear partial differential equations. The nonlinearity creates products involving Heaviside functions and derivatives of the Dirac \(\delta\) distribution and these multiplication problems are dealt with the noncommutative multiplication theory of B. Fuchssteiner [Stud. Math. 77, 439-453 (1984; Zbl 0543.46021)].
Two simple examples illustrate the implementation of the discontinuous basis functions: the dispersionless Korteweg-de Vries equation \(u_ t+(u+1)u_ x=0\) and the Burgers equation \(u_ t+u\cdot u_ x=\epsilon u_{xx}.\) In both cases the method leads to a system of ordinary differential equations which in turn can be solved sequentially. The accuracy obtained compares well with that achieved by the standard product approximation.
Reviewer: P.Laasonen

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 0543.46021
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fuchssteiner, B., Distribution algebras and elementary shock wave analysis, (Vichnevetsky, R.; Stepleman, R. S., Advances in Computer Methods and Partial Differential Equations-V Proceedings of the Fifth IMACS International Symposium on Computer Methods for Partial Differential Equations (June 19-21, 1984), Lehigh University-Bethlehem: Lehigh University-Bethlehem CA)
[2] Rosinger, E. E., Nonlinear Partial Differential Equations, Sequential and Weak Solutions, Vol. 44 (1980), North Holland Mathematics Studies: North Holland Mathematics Studies Amsterdam · Zbl 0459.65066
[3] Colombeau, J. F., Elementary Introduction to New Generalized Functions, Vol. 113 (1985), North Holland Mathematics Studies: North Holland Mathematics Studies Amsterdam · Zbl 0627.46049
[4] Craven, B. D., Generalized Functions for Applications (1982), University of Melbourne, Preprint · Zbl 0567.46018
[5] Christie, I.; Griffiths, D. F.; Mitchell, A. R.; Sanz-Serna, J. M., Product approximation for non-linear problems in the finite element method, IMA J. Num. Analysis, 1, 253-266 (1981) · Zbl 0469.65072
[6] Steppelev, J., Treatment of discontinuous finite element basis functions as distributions, Beitr. Phys. Atmosph., 55, 1, 43-60 (1982)
[7] Hewitt, E.; Stromberg, K., Real and Abstract Analysis (1969), Springer: Springer New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.