Abstract
Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.
Highlights
To determine approximate solutions to fluid flow problems in three-dimensional geometries is a computationally demanding task
We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain
By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems
Summary
To determine approximate solutions to fluid flow problems in three-dimensional geometries is a computationally demanding task. A natural way to approximate the three-dimensional problem is to use Fourier truncation and, to obtain a fully discrete scheme, compute approximate solutions to a finite number of the two-dimensional problems. An added complexity is that the natural, variational spaces for the Fourier coefficients turn out to be weighted Sobolev spaces, where the weight is either the distance to the symmetry axis, or its inverse We derive these spaces by decomposing (through a change of variables to cylindrical coordinates) the threedimensional norms for the relevant spaces L2(Ω ), L20(Ω ), (H1(Ω )), (H01(Ω )) and (H−1(Ω )) into sums over all wavenumbers. ∙ In Section 5, we derive natural variational spaces for the Fourier coefficients by decomposing the relevant three-dimensional norms into sums over all wavenumbers. Stokes equations. ∙ In Section 3, we recall some basic formulas and state the Stokes problem in cylindrical coordinates. ∙ In Section 4, we use Fourier expansion with respect to the angular variable to reduce the three-dimensional Stokes problem to a countable family of two-dimensional problems. ∙ In Section 5, we derive natural variational spaces for the Fourier coefficients by decomposing the relevant three-dimensional norms into sums over all wavenumbers. ∙ In Section 6, we state variational formulations of the two-dimensional problems and show that these are well-posed. ∙ In Section 7, we introduce two families of anisotropic spaces that we need to analyze the error due to Fourier truncation. ∙ In Section 8, we prove an error estimate due to Fourier truncation
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