Computational & Technology Resources
an online resource for computational,
engineering & technology publications
Civil-Comp Proceedings
ISSN 1759-3433
CCP: 90
PROCEEDINGS OF THE FIRST INTERNATIONAL CONFERENCE ON PARALLEL, DISTRIBUTED AND GRID COMPUTING FOR ENGINEERING
Edited by:
Paper 7

Resolution of Non-Linear Parabolic Periodic Problems using Domain Decomposition

N. Alaa1, N. Idrissi1 and J.R. Roche2

1Department of Mathematics and Informatics, Faculty of Science and Technology Gueliz, Marrakech, Morocco
2Elie Cartan Institute of Nancy, Vandoeuvre lès Nancy, France

Full Bibliographic Reference for this paper
N. Alaa, N. Idrissi, J.R. Roche, "Resolution of Non-Linear Parabolic Periodic Problems using Domain Decomposition", in , (Editors), "Proceedings of the First International Conference on Parallel, Distributed and Grid Computing for Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 7, 2009. doi:10.4203/ccp.90.7
Keywords: domain decomposition, nonlinear elliptic partial differential equations.

Summary
In this paper we prove the existence of weak solutions for some quasi-linear parabolic and periodic problems and present a method to compute a numerical solution. More precisely we consider the following quasi-linear parabolic and periodic equations, referred to as Problem 1. We are particularly interested in situations where the growth of J, the nonlinearity with respect to the gradient, is critical (quadratic).

The main result of the paper shows that if a supersolution of Problem 1 exists, then a weak solution u of Problem 1 exists such that 0 <= u <= w.

In the last section of the paper we present a numerical iterative method to solve Problem 1. Formally the iterative method constructs a sequence of numerical solutions of the Yosida approximation of Problem 1, with a first estimate which is a supersolution of the considered problem [1].

The algorithm to compute the supersolution is based on the Schwarz overlapping domain decomposition method, combined with the finite element method. The domain partition should be determined by the behavior of the non-linearity F(t,x,u). In a second step a weak solution of the non-linear problem is computed using a Newton method.

References
1
N. Alaa, J.R. Roche, "Theoretical and Numerical Analysis of a Class of Nonlinear Elliptic Equations", Mediterr. J. Math, 2, 327-344, 2005. doi:10.1007/s00009-005-0048-4

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £72 +P&P)