Simulating Two Phase Immiscible Flow In Heterogeneous Media Using Finite Elements And Multigrid Methods

Streamline upwind/Petrov–Galerkin formulation for convec-tion dominated flows with particular emphasis on the incompressible Navier–Stokes equations. A dynamic mesh partition algorithm for the finite element solution of two phase immiscible flow in oil reservoirs. Finite elements in fluids. K. Characteristic adaptive subdo-main methods for reservoir flow problems. Matrix–dependent pro-longations and restrictions in a blackbox multigrid solver. A new finite element formulation for computational fluid dynamics: IV a discontinuity capturing operator for multidimensional advective diffusive systems. Parallel implementation and performance of multigrid algorithms for solving eigenvalue problems. A parallel multi-grid method for history–dependent elastoplasticity computations. A multi–element group preconditioned GMRES algorithm for nonsymmetric systems arising in finite element analysis. mesh number cycles per nonlinear iteration gm pcg mg 1 2 3 4 0 10 5 Figure 12: Equation Solver Performance for the Homogeneous Five Spot (gm=GMRES, pcg=precondi-tioned conjugate gradient, mg=multigrid). lem size grew. However, the number of cycles per iteration required by conjugate gradient increased with the problem size; the number of cycles were found to be proportional to m 1 2 3 4 cycles per nonlinear iteration 0 10 gm pcg mg 20 30 0 mg* Figure 13: Equation Solver Performance for the Heterogeneous Five Spot on Mesh 3 (gm=GMRES, pcg=preconditioned conjugate gradient, mg=multigrid, mg*=modified multigrid). m 1 2 3 4 0 cycles per nonlinear iteration mg mg* 1 2 3 0 Figure 14: Multigrid Performance for the Heterogeneous Five Spot on Mesh 3 (mg=multigrid, mg*=modified multigrid). n 1ń3. Thus it would appear that multigrid is more suitable for large, three dimensional simulations, since the total computational work will be proportional to n. In addition, the element–by–element implementation employed in this study will render such simulations relatively inexpensive. The solution of a series of heterogeneous five spot problems demonstrated that the performance of all of the iterative methods considered can deteriorate as the degree of hetero-geneity is increased. The interpolation scheme developed for heterogeneous problems appeared to reduce this effect on the multigrid algorithm. This scheme offers some promise for unstructured finite element meshes; however, further work is required to determine its robustness. The multi–grid method for the diffusion equation with strongly discontinuous coefficients .