Collocation Trefftz methods for the stokes equations with singularity

This study explores the boundary methods for the two-dimensional homogeneous Stokes equations and investigates the particular solutions (PS) satisfying the Stokes equations. Smooth solutions for the Stokes equations are provided by explicit fundamental solutions (FS) and PS in this study, and singular corner solutions are also provided from linear elastostatics given in Li et al. ([Eng. Anal. Bound. Elem. 34 (2009), 533-648, 2009). A new singularity model with an interior crack is proposed and solved by the collocation Trefftz method (CTM). The proposed method achieves highly accurate solutions with the first leading coefficient having 10 significant digits. These solutions may be used as a benchmark for testing results obtained by other numerical methods. Error bounds are derived for the CTM solutions using the PS. For a general corner, the exponent νk in r νk can only be obtained by numerical solutions of a system of nonlinear algebraic equations. Therefore, the combined method using many FS plus a few singular solutions is inevitable in most applications. For singularity problems, combining a few singular solutions with the FS is an advanced topic and is successfully implemented in Lee et al. (Eng. Anal. Bound. Elem. 24 (2010), 632–654); however, combining a few singular solutions with the smooth PS fails to converge in the first leading coefficient. As a result, the aforementioned method is not applicable to the singularity problems addressed in this article. With the help of particular and singular solutions, the hybrid Trefftz method with Lagrange multipliers can be developed for the Stokes equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2012