The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the $$L^2$$L2 norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

[1]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .

[2]  V. G. Ferreira,et al.  The MAC method , 2008 .

[3]  Kazufumi Ito,et al.  A high order compact MAC finite difference scheme for the Stokes equations: Augmented variable approach , 2008, J. Comput. Phys..

[4]  Endre Süli.  Convergence of finite volume schemes for Poisson's equation on nonuniform meshes , 1991 .

[5]  Peter D. Minev.  Remarks on the links between low-order DG methods and some finite-difference schemes for the Stokes problem , 2008 .

[6]  Yunqing Huang,et al.  Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids , 2013 .

[7]  Shouhong Wang,et al.  Surface Pressure Poisson Equation Formulation of the Primitive Equations: Numerical Schemes , 2003, SIAM J. Numer. Anal..

[8]  Guido Kanschat,et al.  A locally conservative LDG method for the incompressible Navier-Stokes equations , 2004, Math. Comput..

[9]  X. Wu,et al.  Analysis and convergence of the MAC scheme. II. Navier-Stokes equations , 1996, Math. Comput..

[10]  E. Süli,et al.  A convergence analysis of Yee's scheme on nonuniform grids , 1994 .

[11]  Chi-Wang Shu,et al.  Analysis of Optimal Superconvergence of Discontinuous Galerkin Method for Linear Hyperbolic Equations , 2012, SIAM J. Numer. Anal..

[12]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[13]  Zhonghua Qiao,et al.  Superconvergence and Extrapolation Analysis of a Nonconforming Mixed Finite Element Approximation for Time-Harmonic Maxwell’s Equations , 2011, J. Sci. Comput..

[14]  Bertil Gustafsson,et al.  Boundary Conditions and Estimates for the Steady Stokes Equations on Staggered Grids , 2000, J. Sci. Comput..

[15]  A. Weiser,et al.  On convergence of block-centered finite differences for elliptic-problems , 1988 .

[16]  Yunqing Huang,et al.  Time-Domain Finite Element Methods for Maxwell's Equations in Metamaterials , 2012 .

[17]  R. Nicolaides.  Analysis and convergence of the MAC scheme. I : The linear problem , 1992 .

[18]  B. Fornberg,et al.  A compact fourth‐order finite difference scheme for the steady incompressible Navier‐Stokes equations , 1995 .

[19]  Wei Yang,et al.  Superconvergence of mixed finite element approximations to 3-D Maxwell's equations in metamaterials , 2011, J. Comput. Phys..

[20]  Houde Han,et al.  A New Mixed Finite Element Formulation and the MAC Method for the Stokes Equations , 1998 .

[21]  Roger Temam,et al.  A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance , 2007, Numerische Mathematik.

[22]  Vivette Girault,et al.  Finite-element error estimates for the MAC scheme , 1996 .

[23]  Chao Wu,et al.  Superconvergence Analysis for Linear Tetrahedral Edge Elements , 2015, J. Sci. Comput..

[24]  G. Kanschat.  Divergence‐free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme , 2008 .

[25]  V. I. Lebedev,et al.  Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I , 1964 .