Dual-Variable Schwarz Methods for Mixed Finite Elements

Schwarz methods for the mixed finite element discretization of second order elliptic problems are considered. By using an equivalence between mixed methods and conforming spaces first introduced in [13], it is shown that the condition number of the standard additive Schwarz method applied to the dual-variable system grows at worst like O(1+H/delta) in both two and three dimensions and for elements of any order. Here, H is the size of the subdomains, and delta is a measure of the overlap. Numerical results are presented that verify the bound

Domain Decomposition Methods for Nonconforming Finite Element Spaces of Lagrange-Type

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at the same rate as their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions

A Priori Estimates for Mixed Finite Element Approximations of Second Order Hyperbolic Equations with Absorbing Boundary Conditions

. Optimal order L 1 -in-time, L 2 -in-space a priori error estimates are derived for mixed finite element approximations for both displacement and stress for a second order hyperbolic equation with first order absorbing boundary conditions. Continuous-in-time, explicit-in-time, and implicit-intime procedures are formulated and analyzed. Key Words. mixed finite element methods, second order hyperbolic equations, absorbing boundary conditions AMS(MOS) subject classification. 65M12, 65M15 1. Introduction. Let\Omega be a bounded domain in IR n with Lipschitz boundary, @ \Omega\Gamma and unit outward normal . For fixed 0 ! T ! 1, we discuss mixed finite element approximations of the second order hyperbolic equation with first order absorbing boundary conditions: u tt \Gamma r\Delta Aru = f in\Omega \Theta (0; T ); (1) u t + ff(Aru) \Delta = g on @\Omega \Theta (0; T ); (2) u(\Delta; 0) = u0 in\Omega ; (3) u t (\Delta; 0) = u1 in\Omega : (4) A is a symmetric matrix wi..

Parallel Domain Decomposition Method for Mixed Finite Elements for Elliptic Partial Differential Equations

In this paper we develop a parallel domain decomposition method for mixed finite element methods. This algorithm is based on a procedure first formulated by Glowinski and Wheeler for a two subdomain problem. This present work involves extensions of the above method to an arbitrary number of subdomains with an inner product modification and multilevel acceleration. Both Neumann and Dirichlet boundary conditions are treated. Numerical experiments performed on the Intel iPSC/860 Hypercube are presented and indicate that this approach is scalable and fairly insensitive to variation in coefficients

THE FEASIBILITY OF A NEW OPTIMIZATION APPROACH TO MULTIPHASE FLOW \Lambda

Abstract. A new optimization formulation for multiphase flow in porous media is introduced. A locally mass conservative mixed finite element method is used for the spatial discretization. An unconditionally stable, fully implicit time discretization is also used and leads to a coupled system of nonlinear equations that must be solved at each time step. We reformulate this system as a least squares problem with simple bounds involving only one of the phase saturations. Both a Gauss-Newton method and a BFGS secant method are considered as potential solvers for the optimization problem. Each evaluation of the least squares objective function and gradient requires solving two single-phase self-adjoint, linear, uniformly elliptic partial differential equations for which very efficient solution techniques have been developed. However, numerical experiments suggest that the new formulation may not be computationally competitive with existing methods. Key Words. Multiphase flow, mixed methods, cell-centered finite differences 1. Introduction The flow of two immiscible fluids is modeled by a system of nonlinear transient partial differential equations coupled with nonlinear algebraic constitutive relationships. Spatial discretization combined with stable fully implicit time stepping leads to a nonlinear system of algebraic equations that must be solved at every time step. Even though the differential operators involved are self-adjoint, the use of Newton-type solution methods require the solution of a system of nonsymmetric linear equations to calculate each Newton step. The objective of this work is to develop an algorithm that exploits the fact that the operators are self-adjoint and only requires the solution of linear systems that are symmetric and positive definite. Our work is predicated on the observation that, all other things being equal, it is easier to solve symmetric positive definite linear systems than more general nonsymmetric systems. Unfortunately, we will find that in the new formulation all other things are not equal