17,735 research outputs found
Multidomain Spectral Method for the Helically Reduced Wave Equation
We consider the 2+1 and 3+1 scalar wave equations reduced via a helical
Killing field, respectively referred to as the 2-dimensional and 3-dimensional
helically reduced wave equation (HRWE). The HRWE serves as the fundamental
model for the mixed-type PDE arising in the periodic standing wave (PSW)
approximation to binary inspiral. We present a method for solving the equation
based on domain decomposition and spectral approximation. Beyond describing
such a numerical method for solving strictly linear HRWE, we also present
results for a nonlinear scalar model of binary inspiral. The PSW approximation
has already been theoretically and numerically studied in the context of the
post-Minkowskian gravitational field, with numerical simulations carried out
via the "eigenspectral method." Despite its name, the eigenspectral technique
does feature a finite-difference component, and is lower-order accurate. We
intend to apply the numerical method described here to the theoretically
well-developed post-Minkowski PSW formalism with the twin goals of spectral
accuracy and the coordinate flexibility afforded by global spectral
interpolation.Comment: 57 pages, 11 figures, uses elsart.cls. Final version includes
revisions based on referee reports and has two extra figure
Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media
In this paper, we study the unconditional convergence and error estimates of
a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete
scheme for the equations of incompressible miscible flow in porous media. We
prove that the optimal error estimates hold without any time-step
(convergence) condition, while all previous works require certain time-step
condition. Our theoretical results provide a new understanding on commonly-used
linearized schemes for nonlinear parabolic equations. The proof is based on a
splitting of the error function into two parts: the error from the time
discretization of the PDEs and the error from the finite element discretization
of corresponding time-discrete PDEs. The approach used in this paper is
applicable for more general nonlinear parabolic systems and many other
linearized (semi)-implicit time discretizations
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
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
The Periodic Standing-Wave Approximation: Overview and Three Dimensional Scalar Models
The periodic standing-wave method for binary inspiral computes the exact
numerical solution for periodic binary motion with standing gravitational
waves, and uses it as an approximation to slow binary inspiral with outgoing
waves. Important features of this method presented here are: (i) the
mathematical nature of the ``mixed'' partial differential equations to be
solved, (ii) the meaning of standing waves in the method, (iii) computational
difficulties, and (iv) the ``effective linearity'' that ultimately justifies
the approximation. The method is applied to three dimensional nonlinear scalar
model problems, and the numerical results are used to demonstrate extraction of
the outgoing solution from the standing-wave solution, and the role of
effective linearity.Comment: 13 pages RevTeX, 5 figures. New version. A revised form of the
nonlinearity produces better result
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