In the third set of experiments, we want to investigate the performance of the mixed GMsFEM when the medium a mixture of material with a moderate Poisson’s ratio and nearly incompressible material. We take λ(x) = 109 in the red region, and κ(x) = 1 in the blue region. In this case, the red region represents a nearly incompressible material (Poisson’s ratio ν close to 0.5). The weighted L2 errors of
Figure 3.3: Top: σ11, middle: σ22, bottom: σ12, left: fine solution (dim=50401), middle: coarse solution (dim=762), right: coarse solution (dim=1103).
the stress tensor and the L2 error of the displacement are recorded in Table 3.2. We see that the mixed GMsFEM is robust even when the medium has a very high contrast in the Poisson’s ratio. In the last set of experiments, λ(x) = 109 in the red region, and λ(x) = 106 in the blue region. So both red and blue regions are nearly incompressible materials, but the Poisson’s ratio in red region is closer to 0.5. The weighted L2 errors of the stress tensor and the L2 error of the displacement are recorded in Table 3.3. Again, we observe robust performance of the mixed GMsFEM in this case.
Table 3.2: Experiment set 3, high Poisson’s ratio only in red region.
dim(ΣH×UH) Rel. L2err. forσ Rel. L2err. for u
762 0.146322 0.114340 1103 0.076518 0.112587 1444 0.049423 0.112398 1785 0.039442 0.112376 2126 0.029167 0.112365 2467 0.016472 0.112361 2808 0.013938 0.112361 3149 0.011806 0.112361 3490 0.009878 0.112361 3831 0.008924 0.112361
Table 3.3: Experiment set 4, high Poisson’s ratio in both regions.
dim(ΣH×UH) Rel. L2err. forσ Rel. L2err. for u
1103 0.009956 0.085874 1785 0.003292 0.085712 2467 0.001977 0.085699 3149 0.000923 0.085698 3831 0.000185 0.085698
4
MULTILEVEL COARSE SPACE CONSTRUCTION BY
ρAMGE
This section builds upon previous results on numerical upscaling coming from the multiscale finite element approach (cf., [37]) and element based algebraic multi- grid approach (AMGe) (cf. [67]). The major difference between the method in this section and the mixed GMsFEM is that, we are going to enrich both the pressure space and velocity space at the same time. Moreover, we will see that the ρAMGe framework allows the algorithm to be applied recursively, which results in a multi- level method. The algorithm that we are going to discuss in this section is one of the methods described in [52]. The other method in [52] has a more local construction and therefore more suitable for parallelism. In fact, a slight modification of mixed GMsFEM and the method that we are going to describe in detail can also lead to a local construction for both coarse pressure and velocity spaces. But since parallel implementation is not the main focus of this dissertation, we are not going to discuss too much in that direction.
We build upon developments in the areas exploiting the spectral choice of the degrees of freedom (cf. e.g., [10, 17, 18, 56, 40, 41, 36, 11]), originally proposed for symmetric positive definite (s.p.d.) problems coming from H1-conforming finite element discretizations of second order elliptic equations. More recently, building coarse spaces via spectral problems has also been an active research topic in the domain decomposition (DD) community (cf. e.g., [34, 65, 64, 54]), see also, [71, 72].
While in the DD area, the main goal is to design DD solvers that are robust with respect to coefficient variations for broad classes of PDEs, our goal is to construct coarse spaces with guaranteed approximation properties so that they can be used as discretization (upscaling) tool. This is motivated and explained in more details in the survey [68] which deals with the use of appropriate AMG-based coarse spaces as accurate discretization spaces (i.e., as an upscaling tool).
The proposed method is another generalization of mixed multiscale finite el- ement method [20, 2] because our coarse spaces already contain the coarse spaces in those methods. More specifically, if one pressure basis is picked per agglomerate and one velocity basis is picked per coarse face, the same coarse spaces of the mixed multiscale finite element method are obtained. This additional flexibility in the se- lection of the number of coarse degrees of freedom allows to fine tune the trade-off between accuracy and computational cost. In addition, as explained in Section 2, high-cost setup can be justified if these spaces are used multiple times. This is the case for our target applications (reservoir simulation, uncertainty quantification) that require solving coarse discretization problems of the type considered in this section many times, while the setup is performed only once and can be viewed as off-line cost.
The specific objective of the present section is to extend the spectral method, originally designed for s.p.d. problems, to mixed finite element discretizations of second order elliptic equations, which is an important advancement of the exist-
ing numerical upscaling techniques since it is mass-conservative, a desired feature in practical applications such as in porous media flow simulations. The spectral method allows us to discretize and solve the problem at different scales of spatial resolution: by imposing a stronger tolerance when solving the local eigenproblems, we select additional coarse degrees of freedom (dofs) thus improving the approximation prop- erties of the resulting coarse spaces. This feature is an essential component of the methodology developed in [58] (see also earlier results in [62], [57]) to coarsen the en- tire de Rham fine-grid finite element complex of L2-conforming, H(div)-conforming, H(curl)-conforming, and H1-conforming spaces with approximation properties. More specifically, the spectral method allows improving the approximation properties of the resulting coarse de Rham complex and to have coarse spaces at different scales of resolution.
In this section, we focus on the part of the complex that involves the H(div) and L2–conforming spaces needed for upscaling the mixed finite element discretization of second order elliptic problems of our interest. We note that in some earlier works [24, 16], multiscale velocity spaces are constructed and piecewise constant pressure basis functions are used in a two-level setting.