Direct Expansion Method of Boundary Condition for solving 3D elliptic equations with small parameters in the irregular domain

In this article, a new methodology, Direct Expansion Method of Boundary Condition (DEMBC), is developed to solve 3D elliptic equations in the irregular domain. First, the previous Rational Differential Quadrature Method (Rational Spectral Collocation Method in (Berrut et al. 2005) [8]), developed by Berrut et al. (2005) [8], has been generalized to solve 3D elliptic equations. Second, it is showed that Direct Expansion Method of Boundary Condition is capable of handling boundary problems with higher efficiency. Finally, with the help of conformal mapping (Tee and Trefethen, 2006) [9] and domain decomposition method, DEMBC and 3D-RDQM are able solve three kinds of 3D elliptic equations with small parameters in the irregular domain. Numerous test results justify the accuracy and efficiency of our approach.

[1]  H. Du,et al.  Differential quadrature Trefftz method for Poisson-type problems on irregular domains , 2008 .

[2]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[3]  A nondispersive and nondissipative numerical method for first-order linear hyperbolic partial differential equations , 1987 .

[4]  Manoj Kumar,et al.  A collection of computational techniques for solving singular boundary-value problems , 2009, Adv. Eng. Softw..

[5]  Kevin T. Chu,et al.  A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations , 2007, J. Comput. Phys..

[6]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[7]  C. Shu,et al.  Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method , 2000 .

[8]  S. H. Lui,et al.  Spectral domain embedding for elliptic PDEs in complex domains , 2009 .

[9]  H. Mittelmann,et al.  Recent Developments in Barycentric Rational Interpolation , 2005 .

[10]  Víctor M. Pérez-García,et al.  Spectral smoothed boundary methods: The role of external boundary conditions , 2006 .

[11]  S.S.E. Lam,et al.  Application of the differential quadrature method to two-dimensional problems with arbitrary geometry , 1993 .

[12]  Faruk Civan,et al.  Solving multivariable mathematical models by the quadrature and cubature methods , 1994 .

[13]  Richard Pasquetti,et al.  A Spectral Embedding Method Applied to the Advection-Diffusion Equation , 1996 .

[15]  Jean-Paul Berrut,et al.  Exponential convergence of a linear rational interpolant between transformed Chebyshev points , 1999, Math. Comput..

[16]  W. Chen,et al.  The Study on the Nonlinear Computations of the DQ and DC Methods , 1999, ArXiv.

[17]  Christoph Börgers,et al.  Domain imbedding methods for the Stokes equations , 1990 .

[18]  Lloyd N. Trefethen,et al.  A Rational Spectral Collocation Method with Adaptively Transformed Chebyshev Grid Points , 2006, SIAM J. Sci. Comput..

[19]  Q. Qin,et al.  A meshless method for generalized linear or nonlinear Poisson-type problems , 2006 .

[20]  Xionghua Wu,et al.  Differential quadrature domain decomposition method for problems on a triangular domain , 2005 .

[21]  W. Chen,et al.  The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates , 1999, ArXiv.

[22]  Jean-Paul Berrut,et al.  The errors in calculating the pseudospectral differentiation matrices for C̆ebys̆ev-Gauss-Lobatto points , 1999 .

[23]  C. Bert,et al.  IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER‐ORDER PDEs: APPLICATION TO FREE VIBRATION OF PLATES , 1996 .

[24]  W. Chen,et al.  A Lyapunov Formulation for Efficient Solution of the Poisson and Convection-Diffusion Equations by the Differential Quadrature Method , 2002, ArXiv.