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E. there exists M such that | a(u, ν) \ < Ai II u ||„,, II ν H^n for all u, ν Ε / / ™ ( Ω ) . Ω 30 TOPICS IN NUMERICAL ANALYSIS Lemma 1 Let Γ CC Λ C Ω , b(u, υ) = (ti, Σ , „ b D v) where b Ε C„(Sl), f eH _ ( / Ε / / ( Λ ) if k + δ — 2m < 0), ω EH (Çl) satisfying the relation α ( ω , φ) = ο b{fy ψ) W> €Ξ Coo (Λ). 7/ien exist coefficients c depending on b, Γ and Λ swcA that s < 6 0 s s k+8 2m m k |Ι*,Γ < ^ ( | | ^ ||o,A + l l / H * + 6 - 2 m , A ) Lemma 1 is straightforward generalization of classical results on interior regularity.
D. Theorem 3. (See [3]) We have under the conditions of Lemma 1 \f(z) - Pn (f;z)\ ύ [«(1) - d(-l)]B {da) â [«(1) - « ( - ! ) ] n (33) G. )] J X 1 [ δ ί ζ ) ] (35) ! ^ . ^ ) ^ ) ! D. 4. On the Calculation of Elliptic Integrals Let ι φ { τ ) [ V(i-« )(i-TV) = a d x ( 3 6 ) where τ is a positive number smaller than one. To calculate (36) we insert in (2) da(x) = (1 - x ) ~ a n d / ( x ) = (1 - r * ) " ' . Then the Gauss-Jacobi abscissae are the zeros of Chebychev polynomials 2 Xkn = cos — 1 / 2 2 2 1 2 π (37) the coefficients of the quadrature formula are λ*„ = - (38) η and the orthonormal polynomials p (da; x) must be replaced by \/2/πΤ (χ) (n = l , 2 , .
DESCLOUX [ 1 0 ] Nitsche, J. and Schatz, A. ( 1 9 7 2 ) . On local approximation properties of L projection o n spline-subspaces, Applicable Analysis 2 , 1 6 1 - 1 6 8 . [ 1 1 ] Nitsche, J. and Schatz, A. ( 1 9 7 4 ) . Interior estimates for Ritz-Galerkin methods, Symposium on Finite Elements and Partial Differential Equations. Madison. [ 1 2 ] Strang, G. ( 1 9 7 2 ) . Approximation in the finite element method, Num. Math. 19, 81-98. [ 1 3 ] T h o m é e , V. and Westergren, B. ( 1 9 6 8 ) . Elliptic difference equations and interior regularity, Num.