What is the best iterative scheme to find the solution of a linear minimization problem in image processing?

Question

I have a linear minimization problem as, $$ \inf_u E(u)=‎ \inf_u \frac{\mu}{2} \int_{\Omega}(u-f)^{2}d\Omega+\frac{c}{2}||u||^{2}-\langle u, g \rangle, $$ where $f$ and $g$ are fixed and c is a positive scalar. Since this minimization problem is linear, I tried as follows, $$ ‎\nabla E(u)=\mu(u-f)+cu-g=0, $$ ‎As a result,‎‎ $$ ‎‎‎u=\frac{1}{\mu+c}(\mu f+g).‎ $$ What I have tried is ofcourse traditional and I need an iterative scheme to find u because it is an image denoising problem and I need to find the denoised image. I appreciate it if someone could please help me how to solve this problem with an iterative scheme and whether my own method can be transformed to an iterative one?