A Mixed Multiscale Spectral Generalized Finite Element Method

Abstract

We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral generalized finite element method (MS-GFEM) and exploits the superior local mass conservation properties of mixed finite elements. Following the MS-GFEM framework, optimal local approximation spaces are built for the velocity field by solving local eigenvalue problems over generalized harmonic spaces. The resulting global velocity space is then enriched suitably to ensure inf-sup stability. We develop the mixed MS-GFEM for both continuous and discrete formulations, with Raviart-Thomas based mixed finite elements underlying the discrete method. Exponential convergence with respect to local degrees of freedom is proven at both the continuous and discrete levels. Numerical results are presented to support the theory and to validate the proposed method.

Figures (6)

• Figure 1: Example 1: $\log_{2}(\textbf{error}_v)$ (left) and $\log_{2}(\textbf{error}_p)$ (right) for different numbers of local bases $n_{loc}$ and oversampling layers $\ell$. The oversampling size is $\ell h$.
• Figure 2: Coefficient $A$ for example 2 (left) and 3 (middle). The $x$-component of $\boldsymbol{u}_h^G$ (right) for example 3 with $m=6, l = 15, n_{loc}=6$.
• Figure 3: Example 2: $\log_{2}(\textbf{error}_v)$ (left) and $\log_{2}(\textbf{error}_p)$ (right) for $m=6$.
• Figure 4: Example 2: $\log_{2}(\textbf{error}_v)$ (left) and $\log_{2}(\textbf{error}_p)$ (right) without enrichment.
• Figure 5: Example 3: $\log_{2}(\textbf{error}_v)$ (left) and $\log_{2}(\textbf{error}_p)$ (right) for $m=6$.

Theorems & Definitions (57)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Proposition 2.5
• proof
• Lemma 2.6
• proof
• Proposition 2.7
• proof