Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.Comment: 39 page

Multiblock modeling of flow in porous media and applications

We investigate modeling flow in porous media in multiblock domain. Mixed finite element methods are used for subdomain discretizations. Physically meaningful boundary conditions are imposed on the non-matching interfaces via mortar finite element spaces. We investigate the pollution effect of nonmatching grids error on the numerical solution away from interfaces. We prove that most of the error in the velocity occurs along the interfaces, and that high accuracy is preserved in the interior of the subdomains. In case of discontinuous coefficients, the pollution from the singularity affects the accuracy in the whole domain. We investigate the upscaling error resulting when fine resolution data is approximated on a very coarse scale. Extending work of Wheeler and Yotov, we incorporate this upscaling error in an a posteriori error estimator for the pressure, velocity and mortar pressure. We employ a non-overlapping domain decomposition method reducing the global system to one that is solved iteratively via a preconditioned conjugate gradient method. This approach is suitable for parallel implementation. The balancing domain decomposition method for mixed finite elements following Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve on each iteration. A coarse solve is used to guarantee consistency and to provide global exchange of information. Quasi-optimal condition number bounds independent of the jump in coefficients are derived. We finally consider multiscale mortar mixed finite element discretizations for single and two phase flows. We show optimal convergence and some superconvergence in the fine scale for the solution and its flux. We also derive efficient and reliable a posteriori error estimators suitable for adaptive mesh refinement. We have incorporated the above methods into a parallel multiblock simulator on unstructured prismatic meshes employing a non-overlapping domain decomposition algorithm and mortar spaces. Numerical experiments are presented confirming all theoretical results

Multiscale Methods for Stochastic Collocation of Mixed Finite Elements for Flow in Porous Media

This thesis contains methods for uncertainty quantification of flow in porous media through stochastic modeling. New parallel algorithms are described for both deterministic and stochastic model problems, and are shown to be computationally more efficient than existing approaches in many cases.First, we present a method that combines a mixed finite element spatial discretization with collocation in stochastic dimensions on a tensor product grid. The governing equations are based on Darcy's Law with stochastic permeability. A known covariance function is used to approximate the log permeability as a truncated Karhunen-Loeve expansion. A priori error analysis is performed and numerically verified.Second, we present a new implementation of a multiscale mortar mixed finite element method. The original algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem. This system is then solved in parallel with an iterative method, requiring the solution to local subdomain problems on every interface iteration. Our modified implementation instead forms a Multiscale Flux Basis consisting of mortar functions that represent individual flux responses for each mortar degree of freedom, on each subdomain independently. We show this approach yields the same solution as the original method, and compare the computational workload with a balancing preconditioner.Third, we extend and combine the previous works as follows. Multiple rock types are modeled as nonstationary media with a sum of Karhunen-Loeve expansions. Very heterogeneous noise is handled via collocation on a sparse grid in high dimensions. Uncertainty quantification is parallelized by coupling a multiscale mortar mixed finite element discretization with stochastic collocation. We give three new algorithms to solve the resulting system. They use the original implementation, a deterministic Multiscale Flux Basis, and a stochastic Multiscale Flux Basis. Multiscale a priori error analysis is performed and numerically verified for single-phase flow. Fourth, we present a concurrent approach that uses the Multiscale Flux Basis as an interface preconditioner. We show the preconditioner significantly reduces the number of interface iterations, and describe how it can be used for stochastic collocation as well as two-phase flow simulations in both fully-implicit and IMPES models

Residual-based a posteriori error estimation for immersed finite element methods

In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contrast

Multi-Resolution Analysis Using Wavelet Basis Conditioned on Homogenization

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed in this work, we are able to numerically compute the approximations of the pressure variables. This algorithm can directly be applied to elliptic problems with discontinuous coefficients.We also implemented Java codes to solve the two dimensional elliptic problems using our methods of solutions. Furthermore, we propose homogenization wavelet reconstruction algorithm, fast transform and the inverse transform algorithms that use the results from the solutions of the local problems and the partial derivatives of the pressure variables to reconstruct the solutions

Solving forward and inverse Helmholtz equations via controllability methods

Waves are useful for probing an unknown medium by illuminating it with a source. To infer the characteristics of the medium from (boundary) measurements, for instance, one typically formulates inverse scattering problems in frequency domain as a PDE-constrained optimization problem. Finding the medium, where the simulated wave field matches the measured (real) wave field, the inverse problem requires the repeated solutions of forward (Helmholtz) problems. Typically, standard numerical methods, e.g. direct solvers or iterative methods, are used to solve the forward problem. However, large-scaled (or high-frequent) scattering problems are known being competitive in computation and storage for standard methods. Moreover, since the optimization problem is severely ill-posed and has a large number of local minima, the inverse problem requires additional regularization akin to minimizing the total variation. Finding a suitable regularization for the inverse problem is critical to tackle the ill-posedness and to reduce the computational cost and storage requirement. In my thesis, we first apply standard methods to forward problems. Then, we consider the controllability method (CM) for solving the forward problem: it instead reformulates the problem in the time domain and seeks the time-harmonic solution of the corresponding wave equation. By iteratively reducing the mismatch between the solution at initial time and after one period with the conjugate gradient (CG) method, the CMCG method greatly speeds up the convergence to the time-harmonic asymptotic limit. Moreover, each conjugate gradient iteration solely relies on standard numerical algorithms, which are inherently parallel and robust against higher frequencies. Based on the original CM, introduced in 1994 by Bristeau et al., for sound-soft scattering problems, we extend the CMCG method to general boundary-value problems governed by the Helmholtz equation. Numerical results not only show the usefulness, robustness, and efficiency of the CMCG method for solving the forward problem, but also demonstrate remarkably accurate solutions. Second, we formulate the PDE-constrained optimization problem governed by the inverse scattering problem to reconstruct the unknown medium. Instead of a grid-based discrete representation combined with standard Tikhonov-type regularization, the unknown medium is projected to a small finite-dimensional subspace, which is iteratively adapted using dynamic thresholding. The adaptive (spectral) space is governed by solving several Poisson-type eigenvalue problems. To tackle the ill-posedness that the Newton-type optimization method converges to a false local minimum, we combine the adaptive spectral inversion (ASI) method with the frequency stepping strategy. Numerical examples illustrate the usefulness of the ASI approach, which not only efficiently and remarkably reduces the dimension of the solution space, but also yields an accurate and robust method