Choice of collocation nodes in high-order schemes for the solution of parabolic equations

We obtain collocation nodes for a quadrangular finite element. They increase the accuracy of a collocation grid algorithm for solving an initial boundary value problem for one-dimensional parabolic second-order equations. When solving boundary value and initial boundary value problems for differential equations a wide range of different methods is used [5, 8, 9, 13, 16], which includes finite difference, projection, spectral methods, finite element methods and so on. By now efficient algorithms of different order of accuracy for problems of mathematical physics have been constructed [8, 17]. Some of these algorithms are based on finite element methods. With the advent of numerical collocation (NC) methods considerable advances have been made in the area of solving boundary value problems for ordinary differential equations (ODEs) by numerical methods. Among these methods the NC method, which was first considered in [4,12,15], turns out to be most efficient. Later it was extensively studied in a lot of works (see, e.g.[2, 3, 18]). The NC method based on splines was realized in the package of programs COLSYS [3] for solving boundary value problems for ODEs. This method also turns out to be quite efficient in developing a program complex for solving a linear problem of stability of incompressible fluid flows [18]. The NC method (NCM) has such characteristic features of the finite element method as a division of a domain of a rather complicated form into subdomains (finite elements), the availability of the canonical element, the application of the projection method. On the other hand, NC methods of solving partial differential equations (PDEs) are a generalization of the NCM to ordinary differential equations. When applying the NC approach to solutions of PDEs a number of new problems as compared to ODEs arise. First, it is a decomposition of the computational domain into subdomains. As to ODEs the decomposition is unique, viz. a decomposition into subintervals. However, there are more possibilities of such decompositions as to PDEs. In this paper we consider only a decomposition into curvilinear quadrangles. Second, consistency conditions for ODEs are imposed on each pair of neighbouring intervals only at a single point and are satisfied exactly. The boundary for PDEs is a line along which consistency conditions must be satisfied. Third, the problem of the optimal (as to error minimization) location of points of collocation and consistency in NCM for PDEs is much more complicated than that for ODEs. When studying theoretically the asymptotic error of the method De Boor and Swartz showed [6] that the nodes of the Gauss quadrature formula are the best * Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia The work was supported by the Russian Foundation for the Basic Research (97-01-00819). 360 \B.P. Kolob v] and Yu.I. Molorodov ones for ODEs. This conclusion is confirmed by numerous calculations. The choice of Markov-Lobatto and Chebyshev nodes as collocation nodes turns out to be sufficiently accurate [4, 7, 19]. This problem for PDEs has been studied only for minimal orders of approximation. As to higher orders of approximation some versions of the collocation node location in a chosen finite element are considered in numerical experiments [16]. 1. POSING THE PROBLEM We consider the initial boundary value problem in the space-time domain QT for the one-dimensional parabolic second-order equation Lu = —ut λa(x,<)urjc + b(x,t)ux + c(z,<)u = d(x,t) x e ( X Q i X k } ; *E[T0,T*] with the linear boundary conditions of a mixed type at the boundary (1.1) and with the initial conditions (1.2) We find the solution of the problem (1.1) by the pseudo-Green version of finite element methods [10,11]. We divide the computational domain QT by mutually exclusive lines with respect to t and with respect to χ into a number of finite space-time elements Qtji with boundary Gj,, (Fig. 1) gt,, = (1.4)