Extending error bound from scalar functions to vector functions

Question

A commonly used results in finite element analysis states that: $$ ||u - u_h||_{q, \Omega} \leq C h^{\min (k+1,s)-q} |u|_{s, \Omega} $$ for 'smooth enough' $u$ and the norm/seminorm is defined as (using multi-index notation): $$ ||v||_{H^s(\Omega)} = \left( \sum_{0 \leq {|\alpha|} \leq s} ||D^\alpha v||^2_{L^2(\Omega)} \right)^{1/2} $$ $$ |v|_{H^s(\Omega)} = \left( \sum_{{|\alpha|}=s} ||D^\alpha v||^2_{L^2(\Omega)} \right)^{1/2} $$ with the spaces defined as: $$ H^1(\Omega) = \{ v \in L^2(\Omega): \nabla v \in (L^2(\Omega)^d \}, $$ where $$ L^2(\Omega) = \{ v: \int_\Omega v^2 d\Omega < \infty \}. $$ I wonder if the inequality at the top can be extended for vector functions? Like for $\mathbf{u} = \left(u_x, u_y, u_z \right)$?

Answer 1

Let $\textbf{u}$ be a vector valued function such that $\textbf{u}:=(u_1,u_2,...,u_N)$ whose pointwise elements are members of $\mathbb{R}$. If a scalar valued function has the norm $||\cdot||$ We equip vector valued functions with the norm $||\textbf{u}||=\sqrt{\sum_{i=1}^N||u_i||^2}$. Therefore, your inequality becomes: \begin{equation} \begin{split} ||\textbf{u}-\textbf{u}_h||_{q,\Omega}&=\sqrt{\sum_{i=1}^N ||u_i-u_{h,i}||_{q,\Omega}^2}\\ &\leq \sqrt{\sum_{i=1}^NC_i^2h^{2(\min(k+1,s)-q)}||u_i||_{s,\Omega}^2}\\ &\leq \sqrt{\sup_iC_i^2h^{2(\min(k+1,s)-q)}\sum_{i=1}^N||u_i||_{s,\Omega}^2}\quad \text{Since $h^{2(\min(k+1,s)-q)}$ is independent of $i$. }\\ &=Ch^{\min(k+1,s)-q}\sqrt{\sum_{i=1}^N||u_i||_{s,\Omega}^2}\quad \text{Taking $C=\sqrt{\sup_iC_i^2}$}\\ &=Ch^{\min(k+1,s)-q}||\textbf{u}||_{s,\Omega} \end{split} \end{equation}