A Nonoverlapping Subdomain Algorithm with Lagrange Multipliers and its Object Oriented Implementation for Interface Problems

نویسندگان

Daoqi Yang

DAOQI YANG

چکیده

A parallel nonoverlapping subdomain Schwarz alternating algorithm with Lagrange multipliers and interface relaxation is proposed for linear elliptic interface problems with discontinuities in the solution, its derivatives, and the coefficients. New features of the algorithm include that Lagrange multipliers are introduced on the interface and that it is used to solve equations with discontinuous solution. These equations have important applications to alloy solidification problems [1] and immiscible flow of fluids with different densities and viscosities and surface tension. They do not fit into the Schwarz preconditioning and Schur complement frameworks since the solution is not in H(Ω), but is piecewise in H(Ω), where Ω is the physical domain on which the differential equations are defined. An expression for the optimal interface relaxation parameters is also given. Numerical experiments are conducted for a piecewise linear triangular finite element discretization in object oriented paradigm using C++. In this implementation, the class for subdomain solvers inherits from the class for grid triangulation and contains type bound procedures for forming stiffness matrices and solving linear systems. From software engineering point of view, features like encapsulation, inheritance, polymorphism and dynamic binding, make such domain decomposition algorithms an ideal application area of object oriented programming. The organization of this work is as follows. In Section 2, the domain decomposition method is described for general elliptic interface problems. In Section 3, a finite element approximation with Lagrange multipliers is considered. Then in Section 4, some implementation issues using object oriented techniques in C++ are discussed. Finally in Section 5, numerical examples are provided to check the performance of the method.

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