A Posteriori Error Estimates And Adaptive Finite Element Methods For Elliptic Equations With Variable Coefficients

Since the late 1970s, adaptive finite element methods(AFEM) has always beenthe popular research fields in scientific and engineering computing. Based on thenormal finite element methods, AFEM is a numerical method which can continuouslyimprove the precision of the approximation solution by a posteriori error estimateand automatic adjustment of the mesh. Because of its computational cost is assmall as possible for the required approximate precision, AFEM is a widely usednumerical method featuring high recognition ability, e?ciency and reliability.For the AFEM, a posteriori error estimate is the key and core. Its role is to cal-culate the errors of the approximate solution on the currently concerned mesh whichdiscretes the solving domain, using finite element solution and some known informa-tion. Moreover, we can get the errors'distribution. The quality of a posteriori errorestimate in?uences the e?ciency and reliability of AFEM. Automatic adjustmentof the mesh means that, according to the principle of equilibrating the error, weshould refine or coarsen some units of the current mesh using the a posteriori errorestimator and its distribution, and then give the mesh of the next step.For elliptic equations with variable coe?cients, we present a new type a posteri-ori error estimate in this paper.Its main idea is as follows. First, we solve the problemon the current mesh T_h by finite element methods and obtain the approximate so-lution u_h; Then we global refine the mesh T_h to obtain the auxiliary mesh T_h/2. Onthe auxiliary mesh, we can rapidly assemble the global sti?ness matrix using theelement sti?ness matrices on T_h. After that, we solve the finite element equation bysome smoothly iterative solver with the interpolation of u_h from the finite elementspace on T_h to that of T_h/2 as initial value. Obtaining the the approximate solutionu_h/2,m, we use the energy norm of u_h-u_h/2,m as the a posteriori error estimate. Wealso propose the corresponding adaptive finite element algorithm based on this errorestimate. For the mesh adjustment, we choose the bisection method which is easyto implement and can guarantee the conformity and shape regularity of the meshes.In practice, the results of various types of numerical experiments we carried outconfirm that the a posteriori error estimate and the corresponding adaptive finiteelement algorithm proposed in this paper are e?ective and feasible.