The Effect of Interpolating the Coefficients in Nonlinear Parabolic Galerkin Procedures

Error estimates are derived for a class of Galerkin methods for a quasilinear parabolic equation. In these Galerkin methods, both continuous and discrete in time, the nonlinear coefficient in the differential equation is interpolated into a finite-dimensional function space in order to compute the integrals involved. Asymptotic error estimates of optimal order are produced. Introduction. In order to use Galerkin methods for parabolic problems, it is necessary to compute large numbers of integrals involving the coefficients in the differential equation. An efficient and practically successful method of approximating these integrals is to interpolate or project the coefficients and evaluate the integrals by formula. It is possible to show, for a rather general collection of approximation schemes, that the resulting approximate solution is essentially as good as if the integrals had been evaluated exactly. These procedures are particularly useful on nonlinear parabolic problems in which we have used a Galerkin-type procedure in the space variables and have discretized the time variable. For these procedures, it is necessary to reform the matrices at every time step; the matrix elements are the integrals referred to above. The effect of this is that much of the computation time is spent forming matrices. Hence, economies in the formation of the matrices have a very important effect on the total cost of the computation. We present here several error estimates for approximations of the solution of a particular nonlinear parabolic problem. In the process of proving these estimates, we develop some approximation theory which may be useful in producing similar estimates for other problems. In Section 1, we illustrate how to handle a very simple, specific example. In Section 2, we define the principal differential problem and present several error estimates under abstract hypotheses on the approximation scheme to be used for the coefficients. In Section 3, we develop examples of function spaces and interpolation methods which satisfy the abstract hypotheses of Section 2. Finally, in Section 4, we state some specific applications of the results of Sections 2 and 3. Received November 9, 1973. AMS (MOS) subject classifications (1970). Primary 65N30; Secondary 35K60. *This research was partially supported by the National Science Foundation Grant NSF GP^2228. Copyright © 1975, American Mathematical Society 360 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PARABOLIC GALERKIN PROCESSES 361 1. An Example. Consider the problem bu/dt (3/o-x)(a(x, u)duldx) = 0, (x, 0 G / x (0, T], (1.1) (dw/o-x)(0, t) = (du/ax)(l, t) = 0, 0tUn+1/2 = (t/„+1 í/„)/Aí„ + 1. The function En + l,2 is a prediction of fn +t,2 and is given by (1.3) En + ^-1/2 + &n;+i + &„ &n; + *tn_x (Un. 1/2 u. 1/2 = < í71+(Aí2/2Aí1)(C/1-í/0), yjj. + i/0), „_3/2)> «>2,