Einar M. Rønquist

An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow

In this paper we present a simple, general methodology for the generation of high-order operator decomposition (“splitting”) techniques for the solution of time-dependent problems arising in ordinary and partial differential equations. The new approach exploits operator integration factors to reduce multiple-operator equations to an associated series of single-operator initial-value subproblems. Two illustrations of the procedure are presented: the first, a second-order method in time applied to velocity-pressure decoupling in the incompressible Stokes problem; the second, a third-order method in time applied to convection-Stokes decoupling in the incompressible Navier-Stokes equations. Critical open questions are briefly described.

Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization.

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Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations

” parameter multidomain certified reduced basis (RB) method for rapid and reliable online evaluation of functional outputs associated with parametrized elliptic partial differential equations. We propose, and provide theoretical justification for, a new procedure for adaptive partition (“

Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries

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Spectral element methods for large scale parallel Navier—Stokes calculations

”-refinement) of the parameter domain into smaller parameter subdomains: we pursue a hierarchical splitting of the parameter (sub)domains based on proximity to judiciously chosen parameter anchor points within each subdomain. Subsequently, we construct individual standard RB approximation spaces (“

A Reduced-Basis Element Method

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Stability of the Parareal Algorithm

”-refinement) over each subdomain. Greedy parameter sampling procedures and a posteriori error estimation play important roles in both the “

A computational procedure for part design

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