Multiscale mass conservative domain decomposition preconditioners for elliptic problems on irregular grids

Abstract

Multiscale methods can in many cases be viewed as special types of domain decomposition preconditioners. The localisation approximations introduced within the multiscale framework are dependent upon both the heterogeneity of the reservoir and the structure of the computational grid. While previous works on multiscale control volume methods have focused on heterogeneous elliptic problems on regular Cartesian grids, we have tested the multiscale control volume formulations on two-dimensional elliptic problems involving heterogeneous media and irregular grid structures. Our study shows that the tangential flow approximation commonly used within multiscale methods is not suited for problems involving rough grids. We present a more robust mass conservative domain decomposition preconditioner for simulating flow in heterogeneous porous media on general grids.

Auxiliary variables for 3D multiscale simulations in heterogeneous porous media

“…However, within an iterative setting, an accurate representation of the boundary conditions only affects the first iteration; after the first iteration, the residual does not represent any actual physics but should be regarded as noise. Thus, the best multiscale approximation does not necessarily give the most efficient preconditioner [7].…”

“…Later, this concept was also applied to multiscale coefficients, in what is termed multiscale finite element and multiscale finite volume methods (see [6] for an introduction). While multiscale numerical methods have shown good properties on academic problems, they often fail to live up to their promise on real problems [7]. By exploiting the link between multiscale numerical methods and domain decomposition (DD), multiscale control volume methods can be framed in an iterative setting which greatly increases the potential for robust implementations [8].…”

“…The problem is difficult already in two spatial dimensions, where the state in the coarse variables must be mapped onto a 1D boundary. Strategies proposed to remedy the situation include oversampling [9,10], utilizing global information [10,11] and using specialized boundary conditions [7,12]. The situation becomes worse in three spatial dimensions, since a mapping to a 2D boundary is needed.…”

Mass Conservative Domain Decomposition for Porous Media Flow

“…Contrary to classical domain decomposition methods, MCDD will produce solutions that are mass conservative at any iteration step, thus it is not necessary to reduce the pressure residual to a very low value before solving transport equations. Various aspects of MCDD have been tested for two-dimensional problems (Kippe, et al, 2008;Sandvin, et al, 2011;Lunati, et al, 2011). However, to formulate multiscale methods for three-dimensional problems has turned out to be considerably more difficult in general, and to our knowledge, no applications of MCDDtype methods within an iterative setting have been reported in three dimensions.…”

“…Both the preceding operators require knowledge about the original geometry of the problem, and can thus be seen as geometric methods. If it is desired to implement multiscale methods strictly algebraically, then it is possible to construct algebraic approximations based on the information in , as was explored in Sandvin, et al, 2011.…”