Stable discretization methods with external approximation schemes

Journal of Applied Mathematics and Stochastic Analysis 8, Number 4, 1995, 405-413 STABLE DISCRETIZATION METHODS WITH EXTERNAL APPROXIMATION SCHEMES RAM U. VERMA International Publications 1206 Coed Drive Orlando, Florida 32826, USA (Received September, 1994; Revised May, 1995) ABSTRACT We investigate the external approximation-solvability of nonlinear equations an upgrade of the external approximation scheme of Schumann and Zeidler [3] in the context of the difference method for quasilinear elliptic differential equations. Key words: External Approximation Scheme, Approximation-Solvability, Difference Method. AMS (MOS) subject classifications: 65J15, 47H17. 1. Introduction Based on the inner approximation schemes of Petryshyn [1, 2] for projection methods, Schumann and Zeidler [3] applied an external approximation scheme to difference method for quasilinear elliptic differential equations. Here we generalize the approximation-solvability of nonlinear operator equations corresponding to an external approximation scheme, which upgrades the external approximation of Schumann and Zeidler. Finally, we consider an application to the abstract generalization. For details on the approximation-solvability, see [1-5]. Next, let r o {X, F, Xn, X*, X, A, W, A n, R n, K n, En} be an external approximation scheme represented by a diagram - w F TE n Xn A X X* (1) $R, K Xn A X* where X,F,X n are real Banach spaces with F reflexive and dim X n < c for all n E N. Here is an extension operator, Ks: Rn: X-*X n is a restriction operator, En: n is a linear continuous operator, and W:X--F is a synchronization operator. All operators Rn, K n and E n are linear and continuous with sup [[ < cx. The < cxz, sup h’n < cxz and sup operator W is linear, continuous and injective. Furthermore, all operators A n are continuous. Xn--F I Rn I I I Xn-.X I En The approximation scheme r0 coincides with the following external approximation scheme of Schumann and Zeidler [3], 7rl-{X,F, Xn, X*,X,A,W, An, Rn, En} when g n is the Printed in the U.S.A. (C)1995 by North Atlantic Science Publishing Company 405 RAM U. VERMA 406 identity: w F A X En X* Rn Xn (2) An Xn ---* and r 0 reduces to the inner approximation schemes of Petryshyn [1, 2] for projection methods when F- X, and W and K n are the identities. Let us recall some definitions for the sake of the completeness. In what follows, the symbols "---," and shall denote the strong convergence and weak convergence, respectively. "" DI.1 (Admissible external approximation scheme): The approximation scheme r 0 is called an admissible external approximation scheme if the following implications should hold" (I1) Compatibility condition: For all x E X, as n---,cx, EnKnRnx--*W(x) in F. Synchronization condition: The weak limits in F of the sequences {EnKnxn} and their subsequences are synchronized, that is, if (I2) En,Kn,xn,w f in F as nc, W(X). D1.2 (Discrete convergence): For a sequence (Zn) of elements with N, (n) is said to convere discreel to n iff then f n for all (Discrete* convergence): For a sequence (x,) of functionals with x n e X n for all the sequence (x,)is said to convere discretely* to E X* n---, iff D1.3 * limC [x*,Zn]Xn -[z*,Z]X holds for all sequences (xn) x n e X n with sup I Xn I < and EnKnxn---W(x in F as 2. External Approximation-Solvability In this section, we consider the unique approximation-solvability of the initial value problem Ax- b, and corresponding discretized problem x X, (3) Stable Discretization Methods with External Approximation Schemes 407 Anx n-b n, x n E X n, n-l,2,..., (4) with respect to the approximation scheme 7r 0 represented by the diagram (1). Theorem 2.1: Suppose that the approximation scheme 7r o represents an admissible external approximation scheme, and the following assumptions hold: (A1) Weak Consistency: For all x X, Ax. A,R,z d* (A2) Stability: For all x, y X n and n > no, I Anx- AnY I X*n > (ll -yllx ), (A3) where # is a suitable gauge function. Approximation of the term b in (3): For each b e X*, there exists a sequence (bn) such that b n d* bforb nX*n and for alln_>no Then the following conditions are equivalent" (C1) Solvability: For each b X*, the equation Ax X, b, x has a solution. Unique approximation-solvability: The equation Ax- b is said to be uniquely approximation-solvable if the following implications h01d" For b x*, the equation Ax- b has a unique solution x X. (i) and all n > n0, the approximate equation (ii) For each b n (C2) X Anx n has a unique solution x n (iii) As n---- (x:) bn X n. d b n d*--*t)==C.Xn---*x and EnKnxn---W(x in F. (C3) A-properness: The operator A:XX* is A-proper with respect to the approximation scheme 7r0, that is, if the following implications hold: --b and sup I An,x n, d* I X, imply the existence of a subsequence (Xn,,) such that d x ,,x and Ax- b. More precisely, the theorem can be expressed as follows: If the approximation scheme r0 is an admissible external approximation scheme with weak consistency and stability, then the equation Ax b, x X, is uniquely approximation-solvable iff A is A-proper. Remark 2.2: Let the assumptions (A1)-(A3) hold. Then we have two different situations for using Theorem 2.1" RAM U. VERMA 408 (S1) (s2) Abstract existence theorems imply the unique approximation-solvability, that is, if the equation Ax- b, x E X, has a solution, i.e., (C1) holds, then, by Theorem 2.1, the equation Ax- b is uniquely approximation-solvable, and A: X--X* is A-proper. A-properness implies the unique approximation-solvability, that is, if we show the Aproperness of A:XX* by a direct argument, then the equation Ax b, x X, by Theorem 2.1, is uniquely approximation-solvable. Corollary 2.3: Theorem 2.1 reduces to the theorem of Schumann and Zeidler [3] when K n i’3 the identity. Before proving Theorem 2.1, we give a lemma, crucial to the proof. Lemma 2.4: Let 7r o be an admissible external approximation scheme. implications hold: te <te*::sttPn I *. I < d* ten* ---O=alim I n* I O, (i) (ii) no --,* Assume sup Proof: (i) Let ten* d* te* such that quence, again denoted by (n), As that Then the following , I I up{[, .]: I . I nI < does not hod, Then there is a subse- I I > n for all n. I 1}, there exists a subsequence, again denoted by (ten), such I n I 1 and[ n, Zn] > n for all n. Since SUpn I En I < o and SUPn I Kn I < oo, we have supn I EnKnten I < oo. Given that F is reflexive, EnKnxn---w f in F as n--,c. The synchronization condition (I2) implies that y- w(). Thus, ten* d--*x* leads to which contradicts (5). d* Let (ii) . ten---0. Since there exists a sequence (ten) with I n I 1 and. I I -[. n, ten]l < for all n. By similar arguments as in the proof of (i), there is a subsequence, again denoted by (ten), such that EnKnXn--*W(te in F as n--oe. Stable Discretization Methods with External Approximation Schemes 409 d Since x..* --,0 this imp (...truncated)