Localization of Sub-problems in the Multiscale Finite Element Method

Question

Background:

I'm trying to understand the Multiscale Finite Element Method and I'm reading Effendiev & Hou (2009) Multiscale Finite Element Method. Suppose I'm working with a poisson equation of the form \begin{align}-\nabla\cdot(k(x)\nabla u)=f \text{ in }\Omega\\ u=g \text{ in } \partial\Omega_1 \\ k(x)\nabla u\cdot n = h \text{ in } \partial\Omega_2\end{align}

The algorithm to solve this pde using Multiscale FEM seems fairly straightforward to implement:

1. Partition the domain $\Omega$ into a triangulation $T_H$ consisting of finite coarse-scale (i.e. large) elements $K_i$ and assign a piece-wise polynomial basis $\Phi_i$ for this scale.

2. Each element $K_i$, we solve for a local fine-scale basis function $\phi_i$ by forcing them to satisfy \begin{align}-\nabla\cdot(k(x)\nabla\phi_i)=0 \text{ in } K_i \\ \phi_i=\Phi_i \text{ on } \partial K_i \end{align}

3. Let the coarse scale solution be a linear combination of the (resolved) fine scale basis functions $u_{coarse}=\sum \alpha_i \phi_i$ and solve the variational form of the original PDE with coarse-scale test functions $\Phi_j$ (i.e. petrov-galerkin method). Thus, we obtain a system of equations $$\sum_i \int_\Omega k(x)\nabla\phi_i\nabla\Phi_j = \int_\Omega f\Phi_j$$ for each $j.$

My Question:

While this algorithm seems easy to implement, I'm struggling trying to understand how step 2 is derived. The authors seem to present this localized problem as a given rather than something derived from theory. While I would really love to have a full account of its derivation (or summary thereof), it may be too much to ask for in this forum. Instead, I'd like to focus my question on two key points:

1. Why solve a laplace equation in each element instead of the poisson equation? How does having zero on the RHS help us resolve the fine scale nature of the basis functions?

2. Why use boundary conditions from the coarse scale basis $\Phi_i$? Why not interpolate them from the BC's given on $\partial\Omega$?

Any help would be greatly appreciated!

Answer 1

Comment regarding your Step 3: the original paper from Hou and Wu uses a Galerkin method. So only the basis $(\phi_i)$ is used. In that case, the matrix remains symmetric. My reply will consider the case of a Galerkin method.

Let $a(\cdot,\cdot)$ be the variational form associated with your problem. I assume that $k$ satisfies $k(x) \geq \kappa > 0$. To simplify the discussion, I will assume homogeneous Dirichlet condition (and no Neumann condition).

The problem becomes: find $u \in H^1_0(\Omega)$ such that $$ a(u,v) = \int_{\Omega} fv, \quad \forall v \in H^1_0(\Omega). $$ Note that $a$ defines an inner-product on $H^1_0(\Omega)$ and that the solution $u$ satisfies also $$ \frac{1}{2} a(u,u) - \int_{\Omega} fu = \inf_{v \in H^1_0(\Omega)} \left[ a(v,v) - \int_{\Omega} fv \right]. $$

A key result is $$ H^1_0(\Omega) = \bigoplus_{i=1}^{I} ``H_0^1(K_i)" \oplus E(\Gamma) $$ where $``H_0^1(K_i)"$ denotes functions from $H^1_0(K_i)$ extended by 0 over $\Omega$. $\Gamma$ denotes the interior skeleton of the mesh, i.e. $$ \Gamma = \left( \bigcup_{i=1}^{I} \partial K_i \right) \backslash \partial \Omega. $$ and $E(\Gamma)$ is the set of energy-minimizing extensions of function in $H^{1/2}_{00}(\Gamma)$, i.e. $$ w \in E(\Gamma) \Leftrightarrow w_{|\partial K_i} \in H^{1/2}(\partial K_i) \mbox{ and } \nabla \cdot ( k(x) \nabla w ) = 0 \mbox{ in } K_i $$ The (previously written) decomposition of $H^1_0(\Omega)$ is an orthogonal decomposition for the inner-product $a$. With this orthogonality, minimizing the energy over $H^1_0(\Omega)$ is equivalent to minimizing the energy over each subspace $H^1_0(K_i)$ and $E(\Gamma)$.

You can show that the projection of $u$ into $H^1_0(K_i)$ are "small". MsFEM usually neglects those components.

The functions $\phi_i$ of MsFEM span a subspace of $E(\Gamma)$. You can show that when $k$ is a constant (or 1), the classical P1 or Q1 finite element basis functions span a subspace for the corresponding $E(\Gamma)$.

Regarding Question 2, to the best of my knowledge, the choice of traces $\Phi_i$ remains a "heuristic" parameter in these methods. When a homogeneous Dirichlet problem is posed, interpolating the boundary condition does not yield anything.