| Popis: |
Variational inequalities have found many applications in applied science. A partial list includes obstacles problems, fluid flow in porous media, management science, traffic network, and financial equilibrium problems. However, solving variational inequalities remain a challenging task as they are often subject to some set of complex constraints, for example the obstacle problem. Domain decomposition methods provide great flexibility to handle these types of problems. In our thesis we consider a general variational inequality, its finite element formulation and its equivalence with linear and quadratic programming. We will then present a non-overlapping domain decomposition formulation for variational inequalities. In our formulation, the original problem is reformulated into two subproblems such that the first problem is a variational inequality in subdomain Ω\(^i\) and the other is a variational equality in the complementary subdomain Ω\(^e\). This new formulation will reduce the computational cost as the variational inequality is solved on a smaller region. However one of the main challenges here is to obtain the global solution of the problem, which is to be coupled through an interface problem. Finally, we validate our method on a two dimensional obstacle problem using quadratic programming. |
