Robust Coarsening in Multiscale PDEs

Approximation Result. Consider an abstract symmetric and continuous bilinear form a(·, ·) : V × V 7→ IR, as well as a collection of linear functionals {fl}l=1 ⊂ V ′, where V ⊂ H and H is a Hilbert space with norm ‖ · ‖. We make the following assumptions on a(·, ·), V , H, ‖ · ‖ and {fl}: A1. a(·, ·) is positive semi-definite and defines a semi-norm | · |a on V , i.e. |v|a = a(v, v) ≥ 0, for all v ∈ V. In addition, for v ∈ V , the expression √ ‖v‖2 + |v|a defines a norm on V . A2. Let cq be a generic constant. For all q ∈ IR there exists a vq ∈ V with fl(vq) = ql, and ‖vq‖ . cq‖q‖l2(IRm). A3. There are two constants ca and cf such that ‖v‖ ≤ ca|v|a + cf ∑m l=1 |fl(v)| , for all v ∈ V. (12) Now, as in the specific case above, define for all v ∈ V ,