A HIGH-ORDER, ANALYTICALLY DIVERGENCE-FREE APPROXIMATION METHOD FOR THE TIME-DEPENDENT STOKES PROBLEM

作者：Keim Christopher^*; Wendland Holger^*

来源：SIAM Journal on Numerical Analysis, 2016, 54(2): 1288-1312.

DOI：10.1137/151006196

• radial basis functions
• divergence-free approximation spaces
• time-dependent PDEs
• approximation orders
• meshfree methods

We develop and analyze a new high-order approximation method for the time-dependent Stokes equation. Our method is based upon a Leray projection, which converts the Stokes equation into a vector-valued heat equation. We then employ meshfree approximation spaces to discretize in space. These discrete approximation spaces use a well-chosen pair of matrix-valued kernels such that, among other things, the velocity is approximated by an analytically divergence-free function. Moreover, the pressure part of the solution is computed simultaneously with the velocity, so that no additional auxiliary problem has to be solved and so that no inf-sup conditions have to be satisfied. Finally, since we use collocation in space, no spatial numerical integration is required. We will give a rigorous analysis of the method in the periodic setting but also point out how the method can be used in more general situations.

• 出版日期2016