Recovery methods for evolution and nonlinear problems

Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and “recovered” versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts.
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\nIn the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection. 
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\nOur theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators.
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\nAn extra novelty is an implementation of a coarsening error “preindicator”, with a complete implementation guide in ALBERTA (versions 1.0–2.0). 
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\nIn the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galërkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an
\nappropriate concept of the “finite element Hessian” based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method
\nis illustrated with computational experiments on linear PDEs in nonvariational form.
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\nWe then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newton’s method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint.
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\nWe also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the Monge–Ampère equation
\nbased on using “finite element convexity” as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation.

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