On the stability of the reduced basis method for Stokes equations in parametrized domains

摘要

We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (ⅰ) the rapid convergence of global reduced basis approximations - Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ⅱ) the off-line/on-line computational procedures decoupling the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate an output of interest - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Particular attention is given (ⅰ) to the pressure treatment of incompressible Stokes problem; (ⅱ) to find an equivalent inf-sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem; (ⅲ) to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions; (ⅳ) to reduce computational costs in order to allow real-time solution of parametrized problem.