# Lower bound for bilinear form in FEM

I'm searching for lower bounds of bilinear forms arising in FEM for elliptic second order PDEs with mixed boundaries.

I did some research and found:

$$\max_{v_{h}\in\mathcal{V}_h(\mathcal{\Omega})}a(v_{h}, v_{h}) \geq Ch^{d-2}\, ,$$

where $$d$$ is the Dimension of the domain, $$C$$ a positive constant and $$h$$ the mesh size.

The bilinear form is defined as $$a(v_h,v_h):= \int_\Omega \nabla v_h A \nabla v_h \, d\Omega$$ and $$A$$ is an elliptic operator.

I found this bound quite often but not a single proof to it. Does anyone know some literature or how the proof works?

• Is $A$ an operator or a matrix? Also, is $\|y\|_2$ the $L_2$ norm? – Wolfgang Bangerth Nov 19 '18 at 18:32
• $A$ is an elliptic operator. The original PDE reads: $-div(A \nabla u) = f$ with mixed boundary conditions. The norm is the Euclidian norm. The inequality appears during the calculation of condition number of the stiffness matrix. It is used to get a lower bound for it. But I can not find a proof for it. – Kerem Nov 19 '18 at 18:51
• Then what are domain and range of $A$? I'm actually pretty sure that you want $A$ to be a symmetric and positive definite $d\times d$ matrix. – Wolfgang Bangerth Nov 19 '18 at 23:48
• I don't think your lower bound makes sense -- what is $y$? As written, the $\max$ is not attained (you can take $y$ arbitrarily close to zero). Should this be $v_h$, too? Are you sure the constant is independent of the choice of $V_h$? – Christian Clason Nov 20 '18 at 7:37
• @ChristianClason you are right. I did a big mistake but fixed it now. Sorry for the circumstance – Kerem Nov 20 '18 at 9:48

I agree that $$A$$ should be a symmetric positive matrix, whose smallest eigenvalue is bounded away from 0. Let $$K_A$$ denote your resulting matrix and $$K$$ denote the FEM matrix for the Laplacian on the same mesh. Then you write $$\max_{ v \neq 0 } \frac{ v^T K_A v }{ v^T v } \geq \lambda_{min, A} \max_{ v \neq 0 } \frac{ v^T K v }{ v^T v } \geq \lambda_{min, A} \max_{ v \neq 0 } \frac{ v^T k_{ele} v }{ v^T v }$$ where $$k_{ele}$$ is one arbitrary element matrix.
Whe largest eigenvalue of one arbitrary element matrix is $$O(h^{d-2})$$.