On nonlinear generalized conjugate gradient methods
Numerische Mathematik
Iterative solution methods
Preconditioning operators and Sobolevgradients for nonlinear elliptic problems
Computers & Mathematics with Applications
Sobolev gradient preconditioning for the electrostatic potential equation
Computers & Mathematics with Applications
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Infinite-dimensional gradient method is constructed for non-linear fourth order elliptic BVPs. Earlier results on uniformly elliptic equations are extended to strong nonlinearity when the growth conditions are only limited by the Hilbert space well-posedness. The obtained method is opposite to the usual way of first discretizing the problem. Namely, the theoretical iteration is executed for the BVP itself on the continuous level in the corresponding Sobolev space, reducing the non-linear BVP to auxiliary linear problems. Thus we obtain a class of numerical methods, in which numerical realization is determined by the method chosen for the auxiliary problems. The biharmonic operator acts as a Sobolev space preconditioner, yielding a fixed ratio of linear convergence of the iteration (i.e. one determined by the original coefficients only, independent of the way of solution of the auxiliary problems), and at the same time reducing computational questions to those for linear problems. A numerical example is given for illustration.