## High-order BEM formulations for strongly nonlinear problems governed by quite general nonlinear differential operators.(English)Zbl 0882.65108

The author uses the technique of homotopy in order to obtain an integral formulation for some general nonlinear boundary value problem corresponding to partial differential equations of elliptic type. Therefore, the method refers to those nonlinear problems whose governing equations as well as boundary conditions do not contain any linear term at all.
Starting from such a very general boundary value problem, the author proposes a homotopy mapping and an iterative algorithm where, at each iterative step, a linear boundary value problem is solved by the classical boundary element method (BEM). This is called a general BEM and the author observes that it contains the traditional BEM; i.e. BEM applied to a differential operator which is a sum of a familiar linear operator and a nonlinear operator; consequently the linear operator could be inverted.
The idea seems to be quite valuable and the author has extended it to parabolic problems (Navier-Stokes equations). However, as he realizes, deeper mathematical insight is necessary to improve and justify this method. The present paper contains, as numerical examples, three nonlinear two-point boundary value problems for second-order differential operators.

### MSC:

 65N38 Boundary element methods for boundary value problems involving PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
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### References:

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