Jiang, Lijian (2008-08). Multiscale numerical methods for partial differential equations using limited global information and their applications. Doctoral Dissertation. Thesis uri icon

abstract

  • In this dissertation we develop, analyze and implement effective numerical methods for multiscale phenomena arising from flows in heterogeneous porous media. The main purpose is to develop innovative numerical and analytical methods that can capture the effect of small scales on the large scales without resolving the small scale details on a coarse computational grid. This research activity is strongly motivated by many important practical applications arising in contaminant transport in heterogeneous porous media, oil reservoir simulations and subsurface characterization. In the work, we investigate three main multiscale numerical methods, i.e., multiscale finite element method, partition of unity method and mixed multiscale finite element method. These methods employ limited single or multiple global information. We apply these numerical methods to partial differential equations (elliptic, parabolic and wave equations) with continuum scales. To compute the solution of partial differential equations on a coarse grid, we define global fields such that the solution smoothly depends on these fields. The global fields typically contain non-local information required for achieving a convergence independent of small scales. We present a rigorous analysis and show that the proposed global multiscale numerical methods converge independent of small scales. In particular, a global mixed multiscale finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The numerical results demonstrate that the global multiscale numerical methods achieve high accuracy.
  • In this dissertation we develop, analyze and implement effective numerical methods
    for multiscale phenomena arising from flows in heterogeneous porous media. The
    main purpose is to develop innovative numerical and analytical methods that can
    capture the effect of small scales on the large scales without resolving the small scale
    details on a coarse computational grid. This research activity is strongly motivated
    by many important practical applications arising in contaminant transport in heterogeneous
    porous media, oil reservoir simulations and subsurface characterization.
    In the work, we investigate three main multiscale numerical methods, i.e., multiscale
    finite element method, partition of unity method and mixed multiscale finite
    element method. These methods employ limited single or multiple global information.
    We apply these numerical methods to partial differential equations (elliptic,
    parabolic and wave equations) with continuum scales. To compute the solution of
    partial differential equations on a coarse grid, we define global fields such that the solution
    smoothly depends on these fields. The global fields typically contain non-local
    information required for achieving a convergence independent of small scales. We
    present a rigorous analysis and show that the proposed global multiscale numerical
    methods converge independent of small scales. In particular, a global mixed multiscale
    finite element method is extensively studied and applied to two-phase flows. We present some numerical results for two-phase simulations on coarse grids. The
    numerical results demonstrate that the global multiscale numerical methods achieve
    high accuracy.

publication date

  • August 2008