NUMERICAL ANALYSIS AND MATHEMATICAL MODELLING BANACH CENTER PUBLICATIONS, VOLUME 9 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 994 ON SINGULAR PERTURBATION OF THE STOKES PROBLEM G. M. KOBEL KOV Department of Mechanics and Mathematics, Moscow University 9899 Moscow, Russia In papers [, ] the perturbation of the Stokes problem was studied. We mean the case when the original Stokes problem for incompressible media is replaced by elasticity theory equations with Poisson ratio ν approximately equal to /. In this case it was proved that u ε u 0, p ε p 0 where ε / ν, (u ε, p ε ) is a solution of the boundary value problem for elasticity theory equations and (u 0, p 0 ) is the solution of the Stokes problem. But in the papers mentioned above only the case ε = const was considered. We will consider the case ε = ε(x). Our technique is different but the results are almost the same. So let us consider the boundary value problem for elasticity theory equations when the Lamé coefficient µ is constant. For the sake of simplicity of presentation we will consider the Dirichlet boundary conditions () µ u + (λ + µ) div u = F, u Ω = 0. Unless otherwise stated, we will assume that µ = const. In this case, similarly to [3], the boundary value problem () can be rewritten in the following form: () u + p = f, αp + div u = 0, u Ω = 0 ; here f = F/µ and α = µ/(λ + µ). It can be easily seen that the boundary value problem () has a more general form than () since it covers the case α = 0 on part of or on the whole domain Ω. 99 Mathematics Subject Classification: Primary 35Q30. The paper is in final form and no version of it will be published elsewhere. [79]
80 G. M. KOBEL KOV (3) Along with () let us consider the perturbed boundary value problem For α = const it was proved that u ε + p ε = f ε, (α + ε)p ε + div u ε = 0, u ε Ω = 0. (4) u ε u W + p p ε cε. Below we will establish the validity of this estimate for the case of variable α. Note that 0 α. If p L then by p we will denote the orthogonal projection of p onto the subspace of functions from the space L which are orthogonal to unity (we will denote this subspace by L /R ). Therefore, we have p = s + p, s = const, (p, ) Ω p dx = 0. Let us prove the uniform boundedness of the functions u ε and p ε. To this end, we take the scalar product of the first equation (3) by u ε and the scalar product of the second equation (3) by p ε and sum up the results; thus we obtain (5) u ε + ((α + ε)p ε, p ε ) = (f ε, u ε ). From the first equation (3) for an arbitrary nonzero vector-function φ we obtain (6) ( p ε, ϕ) φ ( u ε, ϕ) φ We use the Babuška Brezzi inequality from [4]: + (f ε, ϕ) φ u ε + f ε. q L c 0 sup ϕ W (q, div ϕ) ϕ. This inequality and the estimate (6) yield (7) p ε c 0 ( u ε + f ε ). Here and above we use the standard notations q = q L, f = sup ϕ W Since α + ε ε > 0, the equality (5) gives the estimate u ε f ε. From this estimate and the inequality (7) we obtain (ϕ, f) ϕ, ϕ = ϕ. (8) u ε + p ε (c 0 + ) f ε. Finally, since f ε f the inequality (8) implies the existence of a constant c independent of ε such that the following inequality is satisfied: (9) u ε + p ε c f. This means that the norms u ε and p ε are uniformly bounded with respect to ε.
STOKES PROBLEM 8 Let p ε = s ε + p ε. Let us prove the uniform boundedness of s ε. Using the expansion of p ε and estimating the right-hand side of the equality (5) by the ε-inequality we obtain (0) 3 4 u ε + s ε(α + ε, ) + s ε ((α + ε), p ε) + ((α + ε)p ε, p ε) c f ε. Therefore, from the estimates (9) and (0) we have () 3 4 u ε + s ε(α + ε, ) + s ε (α + ε, p ε) + ((α + ε)p ε, p ε) + p ε c f. Using the ε-inequality we can estimate the term s ε (α + ε, p ε) on the left-hand side of () as follows: s ε (α + ε, p ε) = (s ε α + ε, α + εp ε ) ε s ε(α + ε, ) + ε ((α + ε)p ε, p ε). Let ε ; substituting this inequality into the estimate () we obtain () 3 ) 4 u ε + ( ε )s ε(α, ) + ( ε ((α + ε)p ε, p ε) + p ε c f. Since we are mainly interested in the asymptotic behaviour as ε 0, we assume that ε. Taking into account that α and choosing ε = 4/5 we obtain from the inequality () the estimate 5 s ε(α, ) + p ε c f. But the inequality (9) yields the boundedness of the norm p ε and thus c s ε f. (α, ) This gives the estimate of the form (3) p ε = s ε + p ε c f. Now we pass to the proof of the estimate (4). Let q ε = p p ε, v ε = u u ε. The errors v ε and q ε are found by solving the problem (4) v ε + q ε = f f ε, αq ε + div v ε = εp ε, v ε Ω = 0. Let us form the scalar product of the first equation (4) and v ε and the scalar product of the second equation (4) and q ε. Summing up the results we obtain the equality (5) v ε + (αq ε, q ε ) = (f f ε, v ε ) + ε(p ε, q ε ). From the first equation (4) we obtain as above the following estimate: q ε c 0 ( v ε + f f ε ). Let us multiply both sides of the last inequality by an arbitrary constant λ and take squares. Summing up this result and the equality (5) and estimating the
8 G. M. KOBEL KOV right-hand side, we have ( ) (6) c 0λ v ε + (αq ε, q ε ) + λ q ε εδ q ε + ε ( ) 4δ p ε + c 0λ 0 f f ε. Let q ε = l ε + q ε and l ε = const. Then the following sequence of relations holds: (αq ε, q ε ) + λ q ε = l ε(α, ) + l ε (α, q ε) + (αq ε, q ε) + λ q ε lε(α, ) + α(q ε, q ε) + (λq ε, q ε) ε lε (α, ) + (αq ε ε, q ε) ) ( ε )lε(α, ) + (λ + ε q ε. Let ε = /( + λ); then the last inequality yields the estimate where (αq ε, q ε ) + λ q ε λ(α, ) + λ l ε + λ q ε c 3 q ε { λ(α, ) c 3 = c 3 (λ) = min + λ, λ }. Note that c > 0 since (α, ) > 0 by assumption. Choose λ = /(8c 0) and δ = c 3 /(ε). Then from the inequality (6) we obtain the estimate (7) 4 v ε + c 3 q ε ε c 3 p ε + c 4 f f ε. This estimate implies the convergences v ε 0 and q ε 0 as ε 0. Moreover, if the f ε converge to f so that the estimate f ε f cε is satisfied then the convergence will be of first order in ε. Thus the estimate (4) is true. So we have proved the following theorem: Theorem. Let the domain Ω, the functions f and f ε and the coefficient α be such that the generalised solutions of the problems () and (3) exist and are unique and f ε f 0. Then the solution (u ε, p ε ) of the problem (3) converges to (u, p) with ε 0 where (u, p) is the solution of the problem (). Moreover, if f ε f cε then we can estimate the rate of convergence of (u ε, p ε ) to (u, p). Namely, in this case u ε u + p ε p cε. R e m a r k. The convergence of (u ε, p ε ) to (u, p) holds true not only in the continuous case but also in the discrete case when the boundary value problems () and (3) are approximated by a finite element method or by a finite difference one. Here it is necessary that the discrete analogue of the Babuška Brezzi inequality is valid while the operators and div are approximated so that ( h ) = div h.
STOKES PROBLEM 83 References [] R. T e m a m, Navier Stokes Equations, North-Holland, 979. [] G. M. Kobel kov, On equivalent norms of subspaces of L, Anal. Math. 3 (977) (in Russian). [3], On an iterative method for solution of the difference equations of elasticity theory, Dokl. Akad. Nauk SSSR 33 (5) (977) (in Russian). [4] G. M. Kobel kov and V. D. Valedinskiĭ, On the inequality p L c p and its finite dimensional image, Soviet J. Numer. Anal. Math. Modelling (3) (986).