New Optimization Approach to Multiphase Flow

A new optimization formulation for simulating multiphase flow in porous media is introduced. A locally mass-conservative, mixed finite-element method is employed for the spatial discretization. An unconditionally stable, fully-implicit time discretization is 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 quasi-Newton 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

Mixed Finite Element Methods on Non-Matching Multiblock Grids

. We consider mixed finite element methods for second order elliptic equations on non-matching multiblock grids. A mortar finite element space is introduced on the non-matching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution, and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory. Key words. Mixed finite element, mortar finite element, error estimates, superconvergence, multiblock, non-conforming grids AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30 1. Introduction. Mixed finite element methods have become popular due to their local (mass) conservation property and ..

Numerical Simulation of Free Surface MHD Flows: Richtmyer-Meshkov Instability and Applications

Numerical methods for free surface MHD ows have been developed and numerical simulations of the Richtmyer - Meshkov type instability in liquid jets in strong magnetic elds caused by an external energy deposition have been performed. Numerical results shed light on the evolution of the proposed Muon Collider target which will be designed as a pulsed jet of mercury interacting with strong proton beams in a 20 Tesla magnetic eld. The formation and evolution of strong instabilities of the jet surface in the absence of the magnetic eld have been demonstrated. We have shown that a uniform magnetic eld signi cantly reduces amplitudes and velocities of surface instabilities and is able to stabilize the jet for all surface perturbations during times typical for the jet breakup at zero magnetic eld