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Using standard rules for diﬀerentiation of distributions (see ) from (5) we obtain ◦ the following weak form of BVP (3), (4): ﬁnd u ∈W21(Ω) such that ◦ ∀ v ∈W21(Ω) , a(u, v) = f, v , (6) where 2 2 a(u, v) = aij Ω i,j=1 ∂u ∂u ∂v ∂v + bi v−u ∂xj ∂xi i=1 ∂xi ∂xi + + cuv dx (7) kuv dΣ . Σ The following assertion is an immediate consequence of the Lax–Milgram lemma  and the imbedding W21 (Ω) ⊂ L2 (Σ). Lemma 1. Let the following assumptions f ∈ W2−1 (Ω), aij , bi , c ∈ L∞ (Ω), 2 aij = aji , c ≥ 0, k ≥ 0, 2 aij (x) zi zj ≥ c0 i,j=1 k ∈ L∞ (Σ), zi2 , c0 = const > 0, ∀ z = (z1 , z2 ) ∈ R2 , a.
G. Stability of Runge-Kutta methods for stiﬀ nonlinear diﬀerential equations, North-Holland, Amsterdam (1984). 2. Dimitriu, G. Parameter identiﬁcation in a two-dimensional parabolic equation using an ADI based solver. In S. Margenov, J. Wasniewski and P. ), Proceedings of The Third International Conference on Large-Scale Scientiﬁc Computing, Lecture Notes in Computer Science, Vol. 2179, (2001) 479-486. , Farago ´ , I. and Havasi, A. 3. Csomo analysis, Comput. Math. , (to appear). ´ On the convergence and local splitting error of diﬀerent ´ , I.
On the construction and comparison of diﬀerence schemes, SIAM J. Num. Anal. 5 (1968) 506-517. 11. Swayne, D. A. Time dependent boundary and interior forcing in locally onedimensional schemes, SIAM Journal on Scientiﬁc and Statistical Computing 8, (1987) 755-767. 12. Yanenko, N. N. The method of fractional steps, Springer, Berlin, (1971). Finite Diﬀerence Approximation of an Elliptic Interface Problem with Variable Coeﬃcients Boˇsko S. Jovanovi´c1 and Lubin G. yu 2 University of Rousse, Department of Mathematics, Studentska str.