A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

Generalized Multiscale Finite Element Methods.Nonlinear Elliptic Equations

In this paper we use the Generalized Multiscale Finite Element Method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast coefficients.The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation.With this convention,we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin(CG)or discontinuous Galerkin(DG)global formulations.Here,we use Symmetric Interior Penalty Discontinuous Galerkin approach.Both methods yield a predictable error decline that depends on the respective coarse space dimension,and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

Time-Lapse 3-D Seismic Wave Simulation via the Generalized Multiscale Finite Element Method

Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs.Repetitive wave modeling using fine layered meshes also adds more computational cost.Conventional approaches such as finite difference and finite element methods may be prohibitively expensive if the whole domain is discretized with the cells corresponding to the grid in the reservoir subdomain.A common approach in this case is to use homogenization techniques to upscale properties of subsurface media and assign the background properties to coarser grid;however,inappropriate application of upscaling might result in a distortion of the model,which hinders accurate monitoring of the fluid change in subsurface.In this work,we instead investigate capabilities of a multiscale method that can deal with fine scale heterogeneities of the reservoir layer and more coarsely meshed rock properties in the surrounding domains in the same fashion.To address the 3-D wave problems,we also demonstrate how the multiscale wave modeling technique can detect the changes caused by fluid movement while the hydrocarbon production activity proceeds.

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.