Finite Element Solution of Boundary Value Problems. Theory by O. Axelsson

By O. Axelsson

Finite point resolution of Boundary worth difficulties: idea and Computation presents a radical, balanced creation to either the theoretical and the computational features of the finite aspect approach for fixing boundary worth difficulties for partial differential equations. even if major advances were made within the finite point procedure considering this ebook first seemed in 1984, the fundamentals have remained an identical, and this vintage, well-written textual content explains those fundamentals and prepares the reader for extra complicated examine. important as either a reference and a textbook, entire with examples and routines, it continues to be as appropriate this day because it used to be whilst initially released

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M - 1, and x w = x. 0, /? l 5 ... is to minimize in each iteration the error ||x — x|| H over a certain subset of RN. 33) and the associated norm ||χ|| Η -ι = (x,x)i;-2i = (xTH-1x)1/\ xeRN. 33) is a well-defined inner product on RN. ) It turns out to be convenient to work with the gradient g = Hx — b rather than directly with x. 36) where ockk) = ( - l) k Π*= Î τ, Φ 0. , Hkg°. Tk is a subset of RN but generally not a subspace. Clearly, gkeTk. 9. 3 21 THE CONJUGATE GRADIENT METHOD Fig. 5. A geometrical interpretation of the best approximation to — g° in the subspace Sk.

1 for an illustration. It is a common feature of the matrices that arise in finite element problems that the envelope is large but only sparsely populated by nonzero entries. 75) expensive with regard to storage and computational labor. If we are willing to settle for an incomplete factorization of H, however, then we can reduce both substantially. 56). , H{N) are restricted to J. There are various ways this modification can be made. Ours is the following: if (i, ])φ J then we put Jig +1) = 0 and add (fcg> - ίίΓΛ#) to the "current value" of h\ï+1).

26), converges to x and the rate of convergence is determined by K(H). 13). 26) the untransformed preconditioned steepest descent method. 25) "transformed" because the sequence y°, y 1 , ... converges to y = £ T x, a transformation of x, rather than to x itself. We can obtain a sequence that converges to x by making the inverse transformation xk = E~Tyk. Naturally, this need not be done in every iteration. Regarding the computational work for the two preconditioned methods, it should first be pointed out that the gradients can be determined economically by recursion.

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