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Approximate approximations. (English) Zbl 0827.65083
Whiteman, J. R. (ed.), The mathematics of finite elements and applications. Highlights 1993. Proceedings of the 8th conference on the mathematics of finite elements and applications, MAFELAP ’93, held at Brunel University, Uxbridge, UK, April 27-30, 1993. Chichester: Wiley. 77-104 (1994).
The starting idea is to represent an arbitrary function as a linear combination of basis functions, which, in contrast to splines, form an approximate partition of unity. The approximations obtained do not converge as the mesh size tends to zero. The lack of the convergence is compensated for, first of all, by the flexibility in the choice of the basis functions and by the simplicity of the generalization to the multidimensional case. Another, and probably the most important, advantage is the possibility to obtain explicit formulae for values of various integral and pseudodifferential operators of mathematical physics applied to the basis functions.
This property, when used for solving elliptic boundary value problems, leads to a new approach to the discretization of boundary integral equations, which is not based upon the decomposition of the boundary into elements. In this approach the coefficients of the resulting algebraic system depend only on the coordinates of a finite number of boundary points; hence the name Boundary Point Method (BPM) seems quite natural.
The first step of the BPM consists of the approximation of the boundary data and of the potential densities by a linear combination of basis functions, each of them being either concentrated near a particular boundary point or decreasing rapidly with the distance from this point. In the second step the calculation of the potentials, whose densities are the basis functions, is carried out. I discuss only one concrete variant of the BPM which leads to an algebraic system with coefficients that are explicitly expressed – up to a simple quadrature.
The approximation method can be employed also to solve various non- stationary problems. Its effectiveness is demonstrated by the numerical solution of Cauchy problems for quasilinear parabolic equations. An important feature of the algorithms is that they are both explicit and stable under much milder restrictions to the time step, depending on the size of the space grid, in comparison with usual explicit difference schemes.
For the entire collection see [Zbl 0818.00016].

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations