Homogenization For Elliptic Finite Difference Equation With Periodic Coefficients

In this paper, we study the difference elliptic equations with periodic coeffcientsand obtain an error estimate between the solution of the homogenized equation andthe origin difference equation. In recent years, many practical applications involvevarious homogenization problems for continue or discrete operators with rapidly os-cillating coeffcients. Classical numerical method for the problem are designed toresolve the full details in fine scale, and the time-consuming is too large to afford. Ifthe coeffcients is of scale-separation properties, we can use the multi-scale methodto get some useful macro information. At present, significant progress has beenachieved in the homogenization of differential equations. There are also a few workfor difference operator, but the main attentions were payed to the homogenizationformula and convergence of discrete systems, only few was payed to the error anal-ysis between the solutions of homogenized equation and the original system, andthey are all related to the stochastic difference equations. To our surprise, there isno error estimates found for the homogenized equation of the difference equationswith periodic coeffcient. Motivated by the above observations, we consider the ho-mogenization of the periodic difference equation and give a detailed error analysis.Similar to the continuous case, by the proof of the existence of antisymmetric matrixfor solenoidal vector in discrete case, we obtained the error estimate between theoriginal equation and the homogenized equation is of O(ε) in the sense of L²-norm,and of O(ε^1/2) in the W^1,2- norm by adding a first-order correction.