A fast null-space method for the unsteady Stokes equations

Abstract

This work is a first taste of using null-space technique to deal with a basic model of unsteady partial differential equations (PDEs), i.e., the unsteady Stokes equations. Any inf–sup stable semi-discretization of the unsteady Stokes equations yields a system of differential–algebraic equations (DAEs), i.e., the unsteady discrete Stokes equations. The exact solution to the unsteady discrete Stokes equations is explicitly constructed with the help of the null-space of the discrete divergence operator and a matrix exponential. In practical implementation, the explicit use of the null-space of the discrete divergence operator is avoided. Therefore, the main workload of solving the unsteady discrete Stokes equations is the matrix exponential vector product. The matrix exponential vector product is written in terms of an integral relation, which is approximated by a linear combination of values of the integrand evaluated at a select number of complex numbers. Each evaluation of the integrand needs to solve a linear system. In numerical experiments, the null-space method outperforms frequently used time-stepping methods in terms of accuracy and computing time.