Seminar in Numerical Analysis: Omar Lakkis (University of Sussex)
Least-squares finite element recovery-based methods provide a simple and practical way to approximate linear elliptic PDEs in nondivergence form where standard variational approach either fails or requires technically complex modifications.
This idea allows the creation of efficient solvers for fully nonlinear elliptic equations, the linearization of which leaves us with an equation in nondivergence form. An important class of fully nonlinear elliptic PDEs can be written in Hamilton--Jacobi--Bellman (Dynamic Programming) form, i.e., as the supremum of a collection of linear operators acting on the unkown.
The least-squares FEM approach, a variant of the nonvariational finite element method, is based on gradient or Hessian recovery and allows the use of FEMs of arbitrary degree. The price to pay for using higher order FEMs is the loss of discrete-level monotonicity (maximum principle), which is valid for the PDE and crucial in proving the convergence of many degree one FEM and finite difference schemes.
Suitable functional spaces and penalties in the least-squares's cost functional must be carefully crafted in order to ensure stability and convergence of the scheme with a good approximation of the gradient (or Hessian) under the Cordes condition on the family of linear operators being optimized.
Furthermore, the nonlinear operator which is not necessarily everywhere differentiable, must be linearized in appropriate functional spaces using semismooth Newton or Howard's policy iteration method. A crucial contribution of our work, is the proof of convergence of the semismooth Newton method at the continuum level, i.e., on infinite dimesional functionals spaces. This allows an easy use of our non-monotone schemes which provides convergence rates as well as a posteriori error estimates.
For further information about the seminar, please visit this webpage.
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