35,162 research outputs found
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
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
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
Multiscale Finite Element Methods for Nonlinear Problems and their Applications
In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the over-sampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities
Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with CordĆØs coefficients
We propose an hp-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman equations with Cordļæ½ĆØs coefficients. The method is proven to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem, and a semismoothness result for the nonlinear operator are also provided
Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes
We consider the lowest-order RaviartāThomas mixed finite element
method for second-order elliptic problems on simplicial meshes in
two and three space dimensions. This method produces saddle-point
problems for scalar and flux unknowns. We show how to easily and
locally eliminate the flux unknowns, which implies the equivalence
between this method and a particular multi-point finite volume
scheme, without any approximate numerical integration. The matrix
of the final linear system is sparse, positive definite for a
large class of problems, but in general nonsymmetric. We next show
that these ideas also apply to mixed and upwind-mixed finite
element discretizations of nonlinear parabolic
convectionādiffusionāreaction problems. Besides the theoretical
relationship between the two methods, the results allow for
important computational savings in the mixed finite element
method, which we finally illustrate on a set of numerical
experiments
A full multigrid method for semilinear elliptic equation
summary:A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term
A Nodally Bound-Preserving Finite Element Method
This work proposes a nonlinear finite element method whose nodal values
preserve bounds known for the exact solution. The discrete problem involves a
nonlinear projection operator mapping arbitrary nodal values into
bound-preserving ones and seeks the numerical solution in the range of this
projection. As the projection is not injective, a stabilisation based upon the
complementary projection is added in order to restore well-posedness. Within
the framework of elliptic problems, the discrete problem may be viewed as a
reformulation of a discrete obstacle problem, incorporating the inequality
constraints through Lipschitz projections. The derivation of the proposed
method is exemplified for linear and nonlinear reaction-diffusion problems.
Near-best approximation results in suitable norms are established. In
particular, we prove that, in the linear case, the numerical solution is the
best approximation in the energy norm among all nodally bound-preserving finite
element functions. A series of numerical experiments for such problems showcase
the good behaviour of the proposed bound-preserving finite element method.Comment: 21 pages, 8 figure
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