Iso-P2 P1/P1/P1 Domain-Decomposition/Finite-Element Method for the Navier-Stokes Equations

In recent years, parallel computers have changed techniques to solve problems in various kinds of fields. In parallel computers of distributed memory type, data can be shared by communication procedures called message-passing, whose speed is slower than that of computations in a processor. From a practical point of view, it is important to reduce the amount of message-passing. Domain-decomposition is an efficient technique to parallelize partial differential equation solvers on such parallel computers. In one type of the domain decomposition method, a Lagrange multiplier for the weak continuity between subdomains is used. This type has the potential to decrease the amount of message-passing since (i) independency of computations in each subdomain is high and (ii) two subdomains which share only one nodal point do not need to execute message-passing each other. For the Navier-Stokes equations, domain decomposition methods using Lagrange multipliers have been proposed. Achdou et al. [1, 2] has applied the mortar element method to the NavierStokes equations of stream function-vorticity formulation. Glowinski et al. [7] has shown the fictitious domain method in which they use the constant element for the Lagrange multiplier. Suzuki [9] has shown a method using the iso-P2 P1 element. But the choice of the basis functions for the Lagrange multipliers has not been well compared in one domain decomposition algorithm. In this paper we propose a domain-decomposition/finite-element method for the Navier-Stokes equations of the velocity-pressure formulation. In the method, subdomain-wise finite element spaces by the iso-P2 P1/P1 elements [3] are used for the velocity and the pressure, respectively. For the upwinding, the upwind finite element approximation based on the choice of upand downwind points [10] is used. For the discretization of the Lagrange multiplier, three cases are compared numerically. As a result, iso-P2 P1/P1/P1 element shows the best accuracy in a test problem. Speed up is attained with the parallelization.