# Best solver/preconditioner for least-squares finite element method

I have seen a lot of literature, lecture videos, etc. on solvers/preconditioners for non-symmetric and/or indefinite systems. However, now I want to solve the mixed poisson/Darcy equation using the least-squares finite element method: $$\int_\Omega\left(\boldsymbol{u}(\mathbf{x}) + \nabla p(\mathbf{x}) - \boldsymbol{g}(\mathbf{x})\right)\cdot \mathbf{A}\left(\boldsymbol{v}(\mathbf{x}) + \nabla q(\mathbf{x})\right)+\left(\nabla\cdot\boldsymbol{u}-f(\mathbf{x})\right)\left(\nabla\cdot\boldsymbol{v}\right)\mathrm{d}\Omega,$$ where $\boldsymbol{u}/p$ and $\boldsymbol{v}/q$ are the velocity/pressure trial and test functions respectively, $\mathbf{A}$ is a symmetric and positive definite weighting tensor, and $\boldsymbol{g}/f$ are the specific body/volumetric source terms respectively. The above weak form will result in a dicrete first order symmetric and positive definite set of equations.

That said, what is the "best" solver/preconditioner for this type of problem? I have tried things like the Conjugate gradient method with Jacobi preconditioning, but it seems the number of iterations increases proportionally with problem size (thus, losing some scalability in the strong sense)

## 1 Answer

Typically multigrid, since least squares FEM usually results in a second order system for which multigrid does well. The convergence of least squares methods is also based on the equivalence of a problem-dependent norm with more standard Sobolev norms, and the same equivalence may sometimes be used to construct appropriate smoothers for multigrid.

Edit: for examples, google searches for FOSLS and multigrid yield some good results. You can also find a good discussion in Gunzberger/Bochev's "Least-Squares Finite Element Methods" book.