13 research outputs found
Optimal L2-error estimates for the semidiscrete Galerkin\ud approximation to a second order linear parabolic initial and\ud boundary value problem with nonsmooth initial data
In this article, we have discussed a priori error estimate for the semidiscrete Galerkin approximation of a general second order parabolic initial and boundary value problem with non-smooth initial data. Our analysis is based on an elementary energy argument without resorting to parabolic duality technique. The proposed technique is also extended to a semidiscrete mixed method for parabolic problems. Optimal L2-error estimate is derived for both cases, when the initial data is in L2
Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data
In this paper, a backward Euler method is discussed for the equations of
motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with
the forcing term independent of time or in in time. It is shown
that the estimates of the discrete solution in Dirichlet norm is bounded
uniformly in time. Optimal a priori error estimate in L2-norm is derived for
the discrete problem with non-smooth initial data. This estimate is shown to be
uniform in time, under the assumption of uniqueness condition
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
We consider the initial/boundary value problem for a diffusion equation
involving multiple time-fractional derivatives on a bounded convex polyhedral
domain. We analyze a space semidiscrete scheme based on the standard Galerkin
finite element method using continuous piecewise linear functions. Nearly
optimal error estimates for both cases of initial data and inhomogeneous term
are derived, which cover both smooth and nonsmooth data. Further we develop a
fully discrete scheme based on a finite difference discretization of the
time-fractional derivatives, and discuss its stability and error estimate.
Extensive numerical experiments for one and two-dimension problems confirm the
convergence rates of the theoretical results.Comment: 22 pages, 4 figure
A Virtual Element Method for a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology
We present a Virtual Element Method (VEM) for a nonlocal reaction-diffusion
system of the cardiac electric field. To this system, we analyze an
-conforming discretization by means of VEM which can make use of
general polygonal meshes. Under standard assumptions on the computational
domain, we establish the convergence of the discrete solution by considering a
series of a priori estimates and by using a general compactness
criterion. Moreover, we obtain optimal order space-time error estimates in the
norm. Finally, we report some numerical tests supporting the theoretical
results
Galerkin approximations for initial value problems with known end time conditions
AbstractGalerkin's method is used to approximate the transient solutions of intial value problems in which a steady state or advanced time state is known. A convergence theorem is established and choices of basis functions are discussed. The method is then applied to systems arising from nuclear reactor kinetics theory and the semi-discretization of parabolic two-point boundary value problems
Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis
Considered here is a class of Boussinesq systems of Nwogu type. Such systems
describe propagation of nonlinear and dispersive water waves of significant
interest such as solitary and tsunami waves. The initial-boundary value problem
on a finite interval for this family of systems is studied both theoretically
and numerically. First, the linearization of a certain generalized Nwogu system
is solved analytically via the unified transform of Fokas. The corresponding
analysis reveals two types of admissible boundary conditions, thereby
suggesting appropriate boundary conditions for the nonlinear Nwogu system on a
finite interval. Then, well-posedness is established, both in the weak and in
the classical sense, for a regularized Nwogu system in the context of an
initial-boundary value problem that describes the dynamics of water waves in a
basin with wall-boundary conditions. In addition, a new modified Galerkin
method is suggested for the numerical discretization of this regularized system
in time, and its convergence is proved along with optimal error estimates.
Finally, numerical experiments illustrating the effect of the boundary
conditions on the reflection of solitary waves by a vertical wall are also
provided
Finite element methods for space-time reactor analysis
"MIT-39-3-5."Also issued as a Sc. D. thesis by Chang Mu Kang in the Dept. of Nuclear Engineering, 1971Includes bibliographical references (leaves 147-150)AT(30-1) 390
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs
Partial differential equations (PDEs) with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension. Tensorized operator equations for spatial and temporal k-point correlation functions of their random solutions are derived. Parametric, deterministic PDEs for the laws of the random solutions are derived. Representations of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos' (GPC) series are established. Recent results on the regularity of solutions of these parametric PDEs are presented. Convergence rates of best N-term approximations, for adaptive stochastic Galerkin and collocation discretizations of the parametric, deterministic PDEs, are established. Sparse tensor products of hierarchical (multi-level) discretizations in physical space (and time), and GPC expansions in parameter space, are shown to converge at rates which are independent of the dimension of the parameter space. A convergence analysis of multi-level Monte Carlo (MLMC) discretizations of PDEs with random coefficients is presented. Sufficient conditions on the random inputs for superiority of sparse tensor discretizations over MLMC discretizations are established for linear elliptic, parabolic and hyperbolic PDEs with random coefficient