By I. Farago, J. Karatson
Contents: MOTIVATION -- Non-linear elliptic equations in version difficulties; Linear algebraic platforms; Linear elliptic difficulties; Non-linear algebraic platforms and preconditioning. THEORETICAL heritage -- Non-linear equations in Hilbert area; Solvability of non-linear elliptic difficulties. ITERATIVE answer OF NON-LINEAR ELLIPTIC BOUNDARY worth difficulties -- Iterative tools in Sobolev house; Preconditioning thoughts for discretise non-linear elliptic difficulties in line with preconditioning operators; Algorithmic realisation of iterative equipment in response to pre-conditioning operators; a few numerical algorithms for non-linear elliptic difficulties in physics; Appendix; Index.
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Additional info for Numerical Solution of Nonlinear Elliptic Problems Via Preconditioning Operators: Theory and Applications (Advances in Computation : Theory and Practice, Volume 11)
7 Weak formulations In this section we give the weak formulations of the main examples which have been so far presented in strong form in this chapter. Since the reader is in general assumed to be familiar with weak formulations, we only go into some details where the symmetry of the weak form requires some calculations. Besides, for the examples not mentioned here the similar weak formulation itself is left to the reader. The weak form of the boundary value problems is obtained in two steps. First, the strong form is transformed such that we multiply it by an arbitrary test function v (taken from the corresponding Sobolev space), we integrate and then apply the divergence theorem to halve the order of derivation in u.
The latter can be obtained from here if we apply the following identities. 1 For any u, v ∈ HD (Ω)3 there holds vol ε(u) · ∇v = vol ε(u) · vol ε(v), dev ε(u) · ∇v = dev ε(u) · dev ε(v). 32 CHAPTER 1. NONLINEAR ELLIPTIC EQUATIONS IN MODEL PROBLEMS Proof. 56) respectively. r. to the product · in the sense that arbitrary matrices A, B ∈ R3×3 satisfy vol A · dev B = 0, and this implies vol A · B = vol A · vol B and dev A · B = dev A · dev B. 56) imply the required identities. 55). 40) by v ∈ H02 (Ω), integration and the divergence theorem.
46) Example Simple iter. Newton Prop. 1 Th. 1 Ths. 7,9,10 Prop. 2 Prop. 3 Prop. 4 Th. 4 Th. 6 Th. 3 Th. 8 Ths. 8,11 Th. 8 Algorithm Sec. 1 Sec. 2 Sec. 4 Sec. 3 Sec. 5– Sec. 6 Table 1. The occurrence of the main examples or corresponding types of problems in the main contexts of the book. The above examples and studied methods form the main line of presentation, as explained in the introduction. 3. 4 together with the dielectric fluid equation. 8 concerning iterations. 4. 34 CHAPTER 1. NONLINEAR ELLIPTIC EQUATIONS IN MODEL PROBLEMS Chapter 2 Linear algebraic systems The aim of this chapter is twofold.