A semi-discrete tailored finite point method for a class of anisotropic diffusion problems

Abstract This work proposes a tailored finite point method (TFPM) for the numerical solution of an anisotropic diffusion problem, which has much smaller diffusion coefficient along one direction than the other on a rectangular domain. The paper includes analysis on the differentiability of the solution to the given problem under some compatibility conditions. It has detailed derivation for a semi-discrete TFPM for the given problem. This work also proves a uniform error estimate on the approximate solution. Numerical results show that the TFPM is accurate as well as efficient for the strongly anisotropic diffusion problem. Examples include those that do not satisfy compatibility and regularity conditions. For the incompatible problems, numerical experiments indicate that the method proposed can still offer good numerical approximations.

[1]  Zhongyi Huang,et al.  A Tailored Finite Point Method for a Singular Perturbation Problem on an Unbounded Domain , 2008, J. Sci. Comput..

[2]  JayanthaPasdunkoraleA..  A SECOND ORDER CONTROL-VOLUME FINITE-ELEMENT LEAST-SQUARES STRATEGY FOR SIMULATING DIFFUSION IN STRONGLY ANISOTROPIC MEDIA , 2005 .

[3]  Joachim Weickert,et al.  Anisotropic diffusion in image processing , 1996 .

[4]  R. Bruce Kellogg,et al.  Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems , 2011, J. Sci. Comput..

[5]  Wenjun Ying,et al.  A kernel-free boundary integral method for elliptic boundary value problems , 2007, J. Comput. Phys..

[6]  M. Shashkov,et al.  Mimetic Finite Difference Methods for Diffusion Equations , 2002 .

[7]  Ian Turner,et al.  A SECOND ORDER CONTROL-VOLUME FINITE-ELEMENT LEAST-SQUARES STRATEGY FOR SIMULATING DIFFUSION IN STRONGLY ANISOTROPIC MEDIA 1) , 2005 .

[8]  S. Valarmathi,et al.  Solving a partially singularly perturbed initial value problem on Shishkin meshes , 2010, Appl. Math. Comput..

[9]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[10]  Zhongyi,et al.  A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM , 2008 .

[11]  R. Bruce Kellogg,et al.  A Tailored Finite Point Method for Convection-Diffusion-Reaction Problems , 2010, J. Sci. Comput..

[12]  Fabrice Deluzet,et al.  An Asymptotic Preserving Scheme for Strongly Anisotropic Elliptic Problems , 2009, Multiscale Model. Simul..

[13]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[14]  Zhongyi Huang,et al.  Tailored finite point method for the interface problem , 2009, Networks Heterog. Media.

[15]  R. B. Kellogg,et al.  Differentiability properties of solutions of the equation -ε 2 δ u + ru = f ( x,y ) in a square , 1990 .

[16]  J. Mullen,et al.  Effect of Bardeen-Herring Correlation on Vacancy Diffusion in Anisotropic Crystals , 1961 .

[17]  Zhongyi Huang,et al.  Tailored finite point method based on exponential bases for convection-diffusion-reaction equation , 2012, Math. Comput..

[18]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[19]  Jitendra Malik,et al.  Scale-Space and Edge Detection Using Anisotropic Diffusion , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  Greg Turk,et al.  Generating textures on arbitrary surfaces using reaction-diffusion , 1991, SIGGRAPH.

[21]  Zhongyi Huang,et al.  Tailored Finite Point Method for a Singular Perturbation Problem with Variable Coefficients in Two Dimensions , 2009, J. Sci. Comput..

[22]  Houde Han,et al.  Tailored finite point method for steady-state reaction-diffusion equations , 2010 .