Charalambos Makridakis

A space-time finite element method for the nonlinear Schro¨dinger equation: the discontinuous Galerkin method

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrodinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in L∞(L 2 ). These estimates are valid under weak restrictions on the space-time mesh.

Implicit-explicit multistep methods for quasilinear parabolic equations

Summary. Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation.

A posteriori error estimates for the Crank–Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T; L2(Ω)) and the higher order spaces, L∞(0, T;H1(Ω)) and H1(0, T; L2(Ω)), with optimal orders of convergence.

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

We approximate the solution of initial boundary value problems for nonlinear parabolic equations. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient, since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels. We derive optimal order error estimates. The abstract results are applied to the Kuramoto-Sivashinsky and the Cahn-Hilliard equations in one dimension, as well as to a class of reaction diffusion equations in R v , v = 2,3.

A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method

The convergence of a class of continuous Galerkin methods for the nonlinear (cubic) Schrodinger equation is analyzed in this paper. These methods allow variable temporal stepsizes as well as changing of the spatial grid from one time level to the next. We show the existence of the resulting approximations and prove optimal order error estimates in

Finite volume schemes for Hamilton-Jacobi equations

L^\infty (L^2 )

A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems

and in

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

L^\infty (H^1 ) .

A posteriori error control for fully discrete Crank–Nicolson schemes

These estimates are valid under weak restrictions on the space-time mesh. These restrictions are milder if the elliptic projection is used at every time step instead of the L2 projection. We also give superconvergence results at the temporal nodes tn.