Electron.n. J. Diff. Eqns., Vol. 1999(1999), No. 18, pp. 1-17.
### On some nonlinear potential problems

M.A. Efendiev, H. Schmitz, & W.L. Wendland

**Abstract:**

The degree theory of mappings is applied to a two-dimensional semilinear
elliptic problem with the Laplacian as principal part subject to a nonlinear
boundary condition of Robin type. Under some growth conditions we
obtain existence. The analysis is based on an equivalent coupled system of
domain--boundary variational equations whose principal parts are the Dirichlet
bilinear
form in the domain and the single layer potential bilinear form on the boundary, respectively.
This system consists of a monotone and a compact part. Additional monotonicity
implies convergence of an appropriate Richardson iteration.
The degree theory also provides the instrument for showing convergence of a
subsequence of a nonlinear finite element - boundary element Galerkin scheme
with decreasing mesh width. Stronger assumptions provide strong monotonicity,
uniqueness and convergence of the discrete Richardson iterations. Numerical
experiments show that the Richardson parameter as well as the number of
iterations (for given accuracy) are independent of the mesh width.
Submitted April 26, 1999. Published May 28, 1999.

Math Subject Classification: 35J65, 47H30, 47H11, 65N30, 65N38.

Key Words: Nonlinear elliptic boundary value problems, degree ofmappings,
finite element - boundary element approximation.

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M. Efendiev

Freie Universitat Berlin, FB Mathematik und Informatik

Arnimallee 2-6, D-14195 Berlin, Germany
H. Schmitz

Sinnersdorfer Str. 173,
D-W-5000 Koln 75, Germany

W. L. Wendland

Universitat Stuttgart, Math. Inst. A

Paffenwaldring 57, D-70569 Stuttgart, Germany

e-mail: wendland@mathematik.uni-stuttgart.de

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