Numerical analysis of semilinear parabolic problems with blow-up solutions.
Numerical computation of soliton dynamics for NLS equations in a driving potential.
Numerical methods with interface estimates for the porous medium equation
Numerical solution of parabolic equations in high dimensions
We consider the numerical solution of diffusion problems in for and for in dimension . We use a wavelet based sparse grid space discretization with mesh-width and order , and discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an -error of for where is the total number of operations,...
Numerical solution of parabolic equations in high dimensions
We consider the numerical solution of diffusion problems in (0,T) x Ω for and for T > 0 in dimension dd ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order pd ≥ 1, and hp discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)-error of O(N-p) for u(x,T)...
Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets
A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. Some numerical example is included to demonstrate the validity and applicability of the technique. The method produces very accurate...
On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation
On a maximum principle for nonlinear equations
On a method for a-posteriori error estimation of approximate solutions to parabolic problems
The aim of the paper is to derive a method for the construction of a-posteriori error estimate to approximate solutions to parabolic initial-boundary value problems. The computation of the suggested error bound requires only the computation of a finite number of systems or linear algebraic equations. These systems can be solved parallelly. It is proved that the suggested a-posteriori error estimate tends to zero if the approximation tends to the true solution.
On a semi-variational method for parabolic equations. I
On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow
On discontinuous Galerkin methods for nonlinear convection-diffusion problems and compressible flow
The paper is concerned with the discontinuous Galerkin finite element method for the numerical solution of nonlinear conservation laws and nonlinear convection-diffusion problems with emphasis on applications to the simulation of compressible flows. We discuss two versions of this method: (a) Finite volume discontinuous Galerkin method, which is a generalization of the combined finite volume—finite element method. Its advantage is the use of only one mesh (in contrast to the combined finite volume—finite...
On error estimates in the Galerkin method for hyperbolic equations.
On finite element approximation of fluid structure interaction by Taylor-Hood and Scott-Vogelius elements
This paper focuses on mathematical modeling and finite element simulation of fluid-structure interaction problems. A simplified problem of two-dimensional incompressible fluid flow interacting with a rigid structure, whose motion is described with one degree of freedom, is considered. The problem is mathematically described and numerically approximated using the finite element method. Two possibilities, namely Taylor-Hood and Scott-Vogelius elements are presented and implemented. Finally, numerical...
On Finite Element Approximations to Time-Dependent Problems.
On -convergence of Rothe’s method
On some Boussinesq systems in two space dimensions: theory and numerical analysis
A three-parameter family of Boussinesq type systems in two space dimensions is considered. These systems approximate the three-dimensional Euler equations, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom. For a subset of these systems it is proved that their Cauchy problem is locally well-posed in suitable Sobolev classes. Further, a class of these systems is discretized...
On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation
We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the norm. We prove optimal order a priori error estimates in the and norms, under mild mesh conditions for two and three space dimensions.
On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation
We discretize the nonlinear Schrödinger equation, with Dirichlet boundary conditions, by a linearly implicit two-step finite element method which conserves the L2 norm. We prove optimal order a priori error estimates in the L2 and H1 norms, under mild mesh conditions for two and three space dimensions.
On the convergence of a multicomponent alternating direction difference scheme.