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.
On the Convergence Rate Estimates for Finite Difference Schemes Approximating Homogeneous Initial-boundary Value Problem for Hyperbolic Equation
On the derivation of the modified equation for the analysis of linear numerical methods
On the strong stability of operator-difference schemes in time-integral norms.
On the Structure of Error Estimates for Finite-Difference Methods.
On the Transport-Diffusion Algorithm and Its Applications to the NavierStokes Equations.
Optimal convergence and a posteriori error analysis of the original DG method for advection-reaction equations
We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order in the -norm if the method uses polynomials of order . Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order . Further we consider a residual-based a posteriori...