On fast solution of the primal biharmonic problem


DOI: 10.34759/trd-2019-108-15

Аuthors

Algazin S. D.1*, Solovyov G. H.2**

1. Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1, prospekt Vernadskogo, Moscow, 119526, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: algazinsd@mail.ru
**e-mail: 19tatarin45@rambler.ru

Abstract

At present, finite element method is the most widespread technique for solving the problems of deformable solid body mechanics. Its shortcomings are well-known and consist in the fact that while displacement approximation by a piecewise-linear function we obtain discontinuous tension. With that, it is worth mentioning that the majority of deformable solid body mechanics problems are described by the equations by the elliptic type equations, which result in smooth solutions. It seems relevant to develop algorithms, which would account for this smoothness. The idea of such algorithms belongs to K.I. Babenko. This idea was put forward by him in the early seventies of the last century. The long-term application of this technique for the elliptic tasks on eigenvalues proved its high efficiency. For example, the task on eigenvalues for the zero equation of Bessel was considered. With this, the first eigenvalue was defined with 22 signs accuracy after the decimal point on the mesh consisting of 22 nodes.

Unlike classical difference methods and finite-element method where dependence of the convergence speed on the number of mesh nodes is of power character, here we have an exponential decrease of error.

The first edition of the book by K.I. Babenko [22] contains the summary of fundamentals of the non-satiable numerical methods. It should be noted, that the studies in computational mathematics in this direction were not sufficiently popularized both in Russia and the world, and they are still practically unknown abroad.

Confirmation to this fact consists in the fact that nowadays an actual “rediscovery” (probably independent) of the same computational techniques started abroad under the name of “spectral” methods (S. Orszag, D. Gotlieb [4], E. Tadmor the USA). It is represented also by (h – p)-specializations of the finite element method (O. Widlund, the USA and S. Schwab, Switzerland), in which the power p of polynomials, employed for functions approximation within one finite element, increases simultaneously with the mesh refinement (i.e. at h → 0). Thus, we can only regret that the works by Babenko and his disciples appeared to be practically forgotten by now.

Keywords:

numerical methods without saturation, h-matrix, circulant

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