103 research outputs found
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 ārecoveredā 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.
In 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.
Our 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.
An extra novelty is an implementation of a coarsening error āpreindicatorā, with a complete implementation guide in ALBERTA (versions 1.0ā2.0).
In 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 Galƫrkin 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
appropriate concept of the āfinite element Hessianā based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method
is illustrated with computational experiments on linear PDEs in nonvariational form.
We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newtonās 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.
We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the MongeāAmpĆØre equation
based on using āfinite element convexityā as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation
A finite element method for second order nonvariational elliptic problems
We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic
problem. The key tools are an appropriate concept of āfinite element Hessianā and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form
A finite element method for second order nonvariational elliptic problems
We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of a 'finite element Hessian' and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasilinear PDE, all in nonvariational form
A finite element method for fully nonlinear elliptic problems
We present a continuous finite element method for some examples of fully
nonlinear elliptic equation. A key tool is the discretisation proposed in
Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of
a linear PDE. An added benefit to making use of this discretisation method is
that a recovered (finite element) Hessian is a biproduct of the solution
process. We build on the linear basis and ultimately construct two different
methodologies for the solution of second order fully nonlinear PDEs. Benchmark
numerical results illustrate the convergence properties of the scheme for some
test problems including the Monge-Amp\`ere equation and Pucci's equation.Comment: 22 pages, 31 figure
Applications of nonvariational finite element methods to Monge--Amp\`ere type equations
The goal of this work is to illustrate the application of the nonvariational
finite element method to a specific Monge--Amp\`ere type nonlinear partial
differential equation. The equation we consider is that of prescribed Gauss
curvature.Comment: 7 pages, 3 figures, tech repor
Discontinuous Galerkin methods for a class of nonvariational problems
We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the ``finite element Hessian'' as an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble, thus this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach.
We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et. al. [2001]. We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the finite element Hessian, thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is the presented showing that the method works well also in a more general setting
A finite element method for fully nonlinear elliptic problems
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems including the Monge-Amp\`ere equation and Pucci's equation
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