Unconditional stability and error estimates of modified characteristics FEMs for the Navier–Stokes equations

• DOI: 10.1007/s00211-015-0767-9
• Publication Date: 2015-10-26T14:35:30Z
• Authors: Zhiyong Si; Jilu Wang; Weiwei Sun
• Keywords: 0101 mathematics; 01 natural sciences

The paper is concerned with the unconditional stability and convergence of characteristics type methods for the time-dependent Navier---Stokes equations. We present optimal error estimates in $$L^2$$L2 and $$H^1$$H1 norms for a typical modified characteristics finite element method unconditionally, while all previous works require certain time-step restrictions. The analysis is based on an iterated characteristic time-discrete system, with which the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the difference between the numerical solution and the solution of the time-discrete system is $$\tau $$?-independent, where $$\tau $$? denotes the time stepsize. Thus numerical solution in $$W^{1,\infty }$$W1,? is bounded and optimal error estimates can be obtained in a traditional way. Numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time-dependent Navier---Stokes equations. The approach used in this paper can be easily extended to many other characteristics-based methods.