## A nonlinear boundary value problem solved by spectral methods.(English)Zbl 0719.35025

We study a nonlinear boundary value problem posed on the interior or the exterior of the unit disk in $${\mathbb{R}}^ 2$$. Using capacity operator, we transform it into a pseudo-differential problem posed on the unit circle. The Galerkin method, with Fourier developments, is used to approximate our problem. We show the convergence of the fixed-point scheme and we give an accurate bound of the $$L^ 2$$-norm of the error. Numerical results coming from a problem arising in electromagnetic casting are also presented.
Reviewer: Olivier Coulaud

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems 65N99 Numerical methods for partial differential equations, boundary value problems 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35S15 Boundary value problems for PDEs with pseudodifferential operators

### Keywords:

capacity operator; Galerkin method
Full Text:

### References:

 [1] DOI: 10.1007/BF00247615 · Zbl 0186.20902 [2] DOI: 10.1007/BF00249500 · Zbl 0242.49029 [3] DOI: 10.1016/S0307-904X(81)80024-8 · Zbl 0475.65078 [4] BREBBIA C.A., Boundary Elements Techniques (1984) · Zbl 0537.73073 [5] CANUTO C., Spectral Methods in Fluid Dynamics (1987) · Zbl 0615.65123 [6] DOI: 10.1137/0519043 · Zbl 0644.35037 [7] Coulaud O., Numerical study of a free boundary problem arising in electromagnetic casting · Zbl 0804.65129 [8] DAUTRAY R., Analyse mathematique et Calcul numerique pour les sciences et les techniques 1 (1984) [9] HENROT A., M2AN 23 pp 155– (1989) [10] HENROT A., About existence of equilibria in electromagnetic cast{$$\neg$$}ing [11] DOI: 10.1016/0021-9991(84)90094-9 · Zbl 0557.65078 [12] DOI: 10.1007/BF01404466 · Zbl 0651.65081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.