Georgios E. Zouraris

On the Construction and Analysis of High Order Locally Conservative Finite Volume-Type Methods for One-Dimensional Elliptic Problems

Locally conservative, finite volume-type methods based on continuous piecewise polynomial functions of degree r \ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension. The new methods extend and generalize the classical finite volume method based on piecewise linear functions. We derive a priori error estimates in the L2, H1, and L^\infty norm and discuss superconvergence effects for the error and its derivative. Explicit, residual-based a posteriori error bounds in the L2 and energy norm are also derived. We compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates.

Adaptive Weak Approximation of Diffusions with Jumps

This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler-Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows and discrete dual backward functions. Combined with proper estimation of the statistical error, they lead to efficient and accurate computation of global error estimates, extending the results by A. Szepessy, R. Tempone, and G. E. Zouraris [Comm. Pure Appl. Math., 54 (2001), pp. 1169-1214]. Adaptive algorithms for either deterministic or trajectory-dependent time stepping are proposed. Numerical examples show the performance of the proposed error approximations and the adaptive schemes.

Convergence rates for adaptive approximation of ordinary differential equations

SummaryThis paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by

A variational principle for adaptive approximation of ordinary differential equations

\mathcal O(N)

Galerkin Methods for Parabolic and Schrödinger Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics

. The alternative version with constant tolerance stops with

Hyperbolic differential equations and adaptive numerics

\mathcal O(N\ {\rm log}\ N)

Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain

total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the

Crank--Nicolson Finite Element Approximations for a Linear Stochastic Fourth Order Equation with Additive Space-Time White Noise

{{L^{{\frac{{1}}{{p+1}}}}}}

A Linear Implicit Finite Difference Discretization of the Schrödinger–Hirota Equation

quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L1-norm of the error density.

An IMEX finite element method for a linearized Cahn–Hilliard–Cook equation driven by the space derivative of a space–time white noise

SummaryA variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error=∑local error⋅weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms.