Constraint energy minimizing generalized multiscale finite element method in the mixed formulation

Abstract

This paper presents a novel mass-conservative mixed multiscale method for solving flow equations in heterogeneous porous media. The media properties (the permeability) contain multiple scales and high contrast. The proposed method solves the flow equation in a mixed formulation on a coarse grid by constructing multiscale basis functions. The resulting velocity field is mass-conservative on the fine grid. Our main goal is to obtain first-order convergence in terms of the mesh size which is independent of local contrast. This is achieved, first, by constructing some auxiliary spaces, which contain global information that cannot be localized, in general. This is built on our previous work on the generalized multiscale finite element method (GMsFEM). In the auxiliary space, multiscale basis functions corresponding to small (contrast-dependent) eigenvalues are selected. These basis functions represent the high-conductivity channels (which connect the boundaries of a coarse block). Next, we solve local problems to construct multiscale basis functions for the velocity field. These local problems are formulated in the oversampled domain, taking into account some constraints with respect to auxiliary spaces. The latter allows fast spatial decay of local solutions and, thus, allows taking smaller oversampled regions. The number of basis functions depends on small eigenvalues of the local spectral problems. Moreover, multiscale pressure basis functions are needed in constructing the velocity space. Our multiscale spaces have a minimal dimension, which is needed to avoid contrast dependence in the convergence. The method’s convergence requires an oversampling of several layers. We present an analysis of our approach. Our numerical results confirm that the convergence rate is first order with respect to the mesh size and independent of the contrast.

Appendices

Appendix

An existence of global and multiscale basis functions

In this appendix, we prove the existence of the problem (??)–(??). It suffices to prove the following. There exists a constant $\widetilde {C}$ such that for all $(u,p)\in V_{0}(K^{+}_{i})\times Q(K^{+}_{i})$, there exist a pair $(v,q)\in V_{0}(K^{+}_{i})\times Q(K^{+}_{i})$ such that

$$\|(u,p)\| \leq \widetilde{C} \, \frac{A((u,p),(v,q))}{\|(v,q)\|} $$

where

$$A((u,p),(v,q)) = a(u,v)-b(v,p)+b(u,q)+s(\pi p,\pi q) $$

and

$$\|(u,p)\|^{2} = \|u\|_{a}^{2} + \|p\|_{s}^{2}. $$

First, it is clear that

$$\|u\|_{a}^{2}+\|\pi p\|_{s}^{2}\leq A((u,p),(u,p)). $$

By the inf-sup condition (??), there exist a w such that ∥w∥_V = ∥(I − π)p∥_s and

$$\|p\|_{s}^{2}\leq C^{-1} \, (\cfrac{b(w,p)}{\|w\|_{V}})^{2}+\|\pi p\|_{s}^{2}. $$

Therefore,

$$\begin{array}{@{}rcl@{}} A((u,p),(w,0)) &\,=\,& a(u,w)\,-\,b(w,p)\!\geq\! a(u,w)\,+\,C \|(I\,-\,\pi)p\|_{s}\|w\|_{V} \\ & \!\geq\!& (C \|(I\,-\,\pi)p\|_{s}\,-\,\|u\|_{a})\|(I\,-\,\pi)p\|_{s} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} A((u,p),\!(0,\tilde{\kappa}^{-1}\nabla\!\cdot\! u)) & \,=\,&s(\pi p,\tilde{\kappa}^{-1}\nabla\cdot u)+\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\cdot u\|_{L^{2}({\Omega})}^{2}\\ & \!\!\geq\!&\! \|\tilde{\kappa}^{-\frac{1}{2}}\nabla\cdot \!u\|_{L^{2}({\Omega})}(\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\!\cdot u\|_{L^{2}({\Omega})}\!\,-\,\!\|\pi p\|_{s}). \end{array} $$

Next, we take $(v,q)=(u+\beta w,p+\tilde {\kappa }^{-1}\nabla \cdot u)$, so

$$\begin{array}{@{}rcl@{}} &&\, \!A((u,p),(v,q)) \\ &&\,=\,\,\|u\|_{a}^{2}+\|\pi p\|_{s}^{2}+\beta(C\|(I-\pi)p\|_{s}-\|u\|_{a})\\&&\!\|(I\!\,-\,\pi)p\|_{s} \,+\,(\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\!\cdot\! u\|_{L^{2}({\Omega})}\!\,-\,\|\pi p\|_{s})\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\!\cdot\! u\|_{L^{2}({\Omega})}\\ &\ &geq\, \|u\|_{a}^{2}+\|\pi p\|_{s}^{2} +\frac{\beta C}{2} \|(I-\pi)p\|_{s}^{2}-\cfrac{\beta}{2C}\|u\|_{a}^{2}\\&&+\frac{1}{2}\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\cdot u\|_{L^{2}({\Omega})}^{2}-\cfrac{1}{2}\|\pi p\|_{s}^{2} \end{array} $$

Finally, we choose β = C, we have

$$\begin{array}{@{}rcl@{}} A((u,p),(v,q))&\!\geq\!&\cfrac{1}{2}\|u\|_{a}^{2}+\cfrac{1}{2}\|\pi p\|_{s}^{2}\\&&\!+\cfrac{1}{2}\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\cdot u\|_{L^{2}({\Omega})}^{2} \!+\cfrac{C^{2}}{2}\|(I-\pi)p\|_{s}^{2} \end{array} $$

and

$$\|(v,q)\|\leq\|(u,p)\|+C\|(I-\pi)p\|_{s}+\|\tilde{\kappa}^{-\frac{1}{2}}\nabla\cdot u\|_{L^{2}({\Omega})}. $$

This completes the proof.