Convergence of Finite Difference Method for Parabolic Problem with Variable Operator
by
Dejan Bojovic
University of Kragujevac, Faculty of Science Radoja Domanovica 12, 34000 Kragujevac, Yugoslavia

We consider the first initial-boundary value problem for parabolic equation with variable coefficients in the domain  Q=\Omega×(0, T]=(0, 1)²×(0, T] ,

\frac\partialu\partialt - 2 å i=1  \frac\partial\partialxi æ è ai(x, t)\frac\partialu\partialxi ö ø =f(x, t) ,  (x, t) in Q , \leqno (1)

u=0 ,  (x, t) in \partial\Omega×[0, T] ,    u(x, 0)=u0(x)  ,  x in \Omega .

We assume that the generalized solution of the problem (1) belongs to the anisotropic Sobolev space  W^{s, s/2}₂(Q) ,  2 < s <= 4 ,   with the right-hand side f(x, t) belonging to  W^{s-2, s/2-1}₂(Q) . Consequently, coefficients  a_i=a_i(x, t)  belong to the space of multipliers M( W^{s-1, (s-1)/2}₂(Q)), i.e. it is sufficient that

ai in Ws-1+\epsilon, (s-1+\epsilon)/24/(s-1)(Q) ,   \epsilon > 0 ,    for  2 < s <= 3 ,

ai in Ws-1, (s-1)/22(Q) ,    for  3 < s <= 4 .

We also assume that the coefficients a_i(x, t) are decreasing functions in variable t.

The initial-boundary value problem (1) is approximated on the uniform mesh Q_h\tau with steps h and \tau by the finite difference scheme with averaged right-hand side. Using Bramble-Hilbert lemma we proved the next result:

Theorem. The difference scheme for the problem (1) converges in the W₂^{2, 1}(Q_{h \tau})   norm, provided   c₁  h² <= \tau <= c₂  h²  . Furthermore,

||u-v||W22, 1(Qh\tau) <= Chs-2 max i  ||ai||Ws-1+\epsilon, (s-1+\epsilon)/24/(s-1)(Q)||u||Ws, s/22(Q) ,  2 < s <= 3

||u-v||W22, 1(Qh\tau) <= Chs-2 max i  ||ai||Ws-1, (s-1)/22(Q)||u||Ws, s/22(Q) ,  3 < s <= 4

These estimates are consistent with the smoothness of the data.

Date received: February 17, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caen-31.