2,079 research outputs found
The Vanishing Moment Method for Fully Nonlinear Second Order Partial Differential Equations: Formulation, Theory, and Numerical Analysis
The vanishing moment method was introduced by the authors in [37] as a
reliable methodology for computing viscosity solutions of fully nonlinear
second order partial differential equations (PDEs), in particular, using
Galerkin-type numerical methods such as finite element methods, spectral
methods, and discontinuous Galerkin methods, a task which has not been
practicable in the past. The crux of the vanishing moment method is the simple
idea of approximating a fully nonlinear second order PDE by a family
(parametrized by a small parameter \vepsi) of quasilinear higher order (in
particular, fourth order) PDEs. The primary objectives of this book are to
present a detailed convergent analysis for the method in the radial symmetric
case and to carry out a comprehensive finite element numerical analysis for the
vanishing moment equations (i.e., the regularized fourth order PDEs). Abstract
methodological and convergence analysis frameworks of conforming finite element
methods and mixed finite element methods are first developed for fully
nonlinear second order PDEs in general settings. The abstract frameworks are
then applied to three prototypical nonlinear equations, namely, the
Monge-Amp\`ere equation, the equation of prescribed Gauss curvature, and the
infinity-Laplacian equation. Numerical experiments are also presented for each
problem to validate the theoretical error estimate results and to gauge the
efficiency of the proposed numerical methods and the vanishing moment
methodology.Comment: 141 pages, 16 figure
Mixed Interior Penalty Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions
This article is concerned with developing efficient discontinuous Galerkin methods for approximating viscosity (and classical) solutions of fully nonlinear second-order elliptic and parabolic partial differential equations (PDEs) including the Monge–Ampère equation and the Hamilton–Jacobi–Bellman equation. A general framework for constructing interior penalty discontinuous Galerkin (IP-DG) methods for these PDEs is presented. The key idea is to introduce multiple discrete Hessians for the viscosity solution as a means to characterize the behavior of the function. The PDE is rewritten in a mixed form composed of a single nonlinear equation paired with a system of linear equations that defines multiple Hessian approximations. To form the single nonlinear equation, the nonlinear PDE operator is replaced by the projection of a numerical operator into the discontinuous Galerkin test space. The numerical operator uses the multiple Hessian approximations to form a numerical moment which fulfills consistency and g-monotonicity requirements of the framework. The numerical moment will be used to design solvers that will be shown to help the IP-DG methods select the “correct” solution that corresponds to the unique viscosity solution. Numerical experiments are also presented to gauge the effectiveness and accuracy of the proposed mixed IP-DG methods
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
Rational Construction of Stochastic Numerical Methods for Molecular Sampling
In this article, we focus on the sampling of the configurational
Gibbs-Boltzmann distribution, that is, the calculation of averages of functions
of the position coordinates of a molecular -body system modelled at constant
temperature. We show how a formal series expansion of the invariant measure of
a Langevin dynamics numerical method can be obtained in a straightforward way
using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics
integrators in terms of their invariant distributions and demonstrate a
superconvergence property (4th order accuracy where only 2nd order would be
expected) of one method in the high friction limit; this method, moreover, can
be reduced to a simple modification of the Euler-Maruyama method for Brownian
dynamics involving a non-Markovian (coloured noise) random process. In the
Brownian dynamics case, 2nd order accuracy of the invariant density is
achieved. All methods considered are efficient for molecular applications
(requiring one force evaluation per timestep) and of a simple form. In fully
resolved (long run) molecular dynamics simulations, for our favoured method, we
observe up to two orders of magnitude improvement in configurational sampling
accuracy for given stepsize with no evident reduction in the size of the
largest usable timestep compared to common alternative methods
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
A modified Galerkin/finite element method for the numerical solution of the Serre-Green-Naghdi system
A new modified Galerkin / Finite Element Method is proposed for the numerical
solution of the fully nonlinear shallow water wave equations. The new numerical
method allows the use of low-order Lagrange finite element spaces, despite the
fact that the system contains third order spatial partial derivatives for the
depth averaged velocity of the fluid. After studying the efficacy and the
conservation properties of the new numerical method, we proceed with the
validation of the new numerical model and boundary conditions by comparing the
numerical solutions with laboratory experiments and with available theoretical
asymptotic results
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