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

Acceleration Methods for Nonlinear Solvers and Application to Fluid Flow Simulations

This thesis studies nonlinear iterative solvers for the simulation of Newtonian and non- Newtonian fluid models with two different approaches: Anderson acceleration (AA), an extrapolation technique that accelerates the convergence rate and improves the robustness of fixed-point iterations schemes, and continuous data assimilation (CDA) which drives the approximate solution towards coarse data measurements or observables by adding a penalty term. We analyze the properties of nonlinear solvers to apply the AA technique. We consider the Picard iteration for the Bingham equation which models the motion of viscoplastic materials, and the classical iterated penalty Picard and Arrow-Hurwicz iterations for the incompressible Navier–Stokes equations (NSE) which model the Newtonian fluid flows. All these nonlinear solvers have some drawbacks. They lack robustness, and the required number of iterations for convergence could be large. In this thesis, we show that AA significantly improves the convergence properties of these nonlinear solvers, makes them more robust, and significantly reduces the number of iterations for convergence. We support our accelerated convergence analysis with various numerical tests. We also consider CDA, which is used to improve the convergence of Picard iterations for the first time in literature. We analyze the improved contraction property of CDA applied to the Picard scheme for steady NSE. We give results for several numerical experiments of CDA applied to the Picard iteration to solve 1D, 2D and 3D nonlinear partial differential equations and show that significant reduction in the required number of iterations thanks to CDA

A sequential regularization method for time-dependent incompressible Navier--Stokes equations

The objective of the paper is to present a method, called sequential regularization method (SRM), for the nonstationary incompressible Navier-Stokes equations from the viewpoint of regularization of differential-algebraic equations (DAEs) , and to provide a way to apply a DAE method to partial differential-algebraic equations (PDAEs). The SRM is a functional iterative procedure. It is proved that its convergence rate is O(ffl m ), where m is the number of the SRM iterations and ffl is the regularization parameter. The discretization and implementation issues of the method are considered. In particular, a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes. It appears that the SRM formulation is new in the Navier-Stokes context. Unlike other regularizations or pseudo-compressibility methods in the Navier-Stokes context, the regularization parameter ffl in the SRM need not be very small, and the regularized..

Numerical approximation of partial differential equations

This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as wel

Efficient solvers for incompressible fluid flows in geosciences

Saddle point problems involving large systems of linear equations arise in a wide variety of applications in computational science and engineering. A variety of solvers have been developed for these type of problems typically with specific applications in mind. In this paper we will focus on saddle point problems as they arise from incompressible fluid flow problems in applications in geosciences. They are characterized through a spatially variable viscosity when modeling temperature dependencies (e.g. in Earth mantel convection models) or moving material interfaces (e.g. in subduction zones simulation and numerical volcano models). In this paper we will give an overview on some of the iterative techniques that can be used and discuss suitable preconditioning techniques. We will discuss the implementation of the schemes using the python module Escript and compare the efficiency of these schemes when applied to convection models on a parallel computer

Interface modeling in incompressible media using level sets in Escript

We use a finite element (FEM) formulation of the level set method to model geological fluid flow problems involving interface propagation. Interface problems are ubiquitous in geophysics. Here we focus on a Rayleigh-Taylor instability, namely mantel plumes evolution, and the growth of lava domes. Both problems require the accurate description of the propagation of an interface between heavy and light materials (plume) or between high viscous lava and low viscous air (lava dome), respectively. The implementation of the models is based on Escript which is a Python module for the solution of partial differential equations (PDEs) using spatial discretization techniques such as FEM. It is designed to describe numerical models in the language of PDEs while using computational components implemented in C and C++ to achieve high performance for time-intensive, numerical calculations. A critical step in the solution geological flow problems is the solution of the velocity-pressure problem. We describe how the Escript module can be used for a high-level implementation of an efficient variant of the well-known Uzawa scheme. We begin with a brief outline of the Escript modules and then present illustrations of its usage for the numerical solutions of the problems mentioned above