Constructive Sobolev gradient preconditioning for semilinear elliptic systems. (English) Zbl 1109.65308

Summary: We present a Sobolev gradient type preconditioning for iterative methods used in solving second order semilinear elliptic systems; the \(n\)-tuple of independent Laplacians acts as a preconditioning operator in Sobolev spaces. The theoretical iteration is done at the continuous level, providing a linearization approach that reduces the original problem to a system of linear Poisson equations. The method obtained preserves linear convergence when a polynomial growth of the lower order reaction type terms is involved. For the proof of linear convergence for systems with mixed boundary conditions, we use suitable energy spaces. We use Sobolev embedding estimates in the construction of the exact algorithm. The numerical implementation has focus on a direct and elementary realization, for which a detailed discussion and some examples are given.


65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
49M15 Newton-type methods
65F10 Iterative numerical methods for linear systems


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