A Posteriori Error Estijmates Based On Gradient Recovery And Adaptive Finite Element Methods

In this thesis we consider the a posteriori error estimation based on gradient recovery andadaptive finite element methods．Superconvergence techniques are very important numerical toolsto achieve high accuracy finite element approximations．As people recognize the advantage of su-perconvergence techniques in the a posteriori error estimation and adaptive finite element methods,people pay more attention to this subject．The well known superconvergence results require that themesh should satisfy some strong conditions,while mesh refinement would make the mesh not nleetthese conditions,and this limits the use of superconvergence techniques in adaptive finite elementmethods．We establish a fundamental estinmtion on general triangular meshes,and we present acomputable parameter to show the relationship between the mesh's qudity and the convergencerate of error．With the basic error expansions,one can verify the superconvergence of the gradientofthe finite elmnent solution u_h and to the gradient ofthe interpolant ui on some types oftriangularmeshes．Numerical tests are presented to illustrate our theoretical results．Finite elenmnt recovery techniques are post-processing methods that reconstruct numericalapproxinmtions from finite elenmnt solutions to obtain improved solutions．The practical usage ofthe recovery technique is not only to improve the quality ofthe approximation,but also to constructa posteriori error estimators in adaptive colnputation．Gradient recovery mchnique is widely usedin engineering practice for its robustness as an a posteriori error estinmtor,its superconvergence ofthe recovered derivatives,and its efficiency in implenmntation．A new gradient recovery techniqueSCR(Superconvergent Cluster Recovery)is proposed and analyzed for finite elenmnt methods．Alinear polynomial approximation is obtained by a least-squares fitting to the finite elenmnt solutionat certain sample points,which in turn gives the recovered gradient at recovering points．The SCRcan be used in a posteriori error esfinmtes,which is relatively simple to implenmnt,cheap in ternlsof storage and colnputational cost for adaptive algorithms．We present some numerical examplesillustrating the effectiveness of our recovery procedure．We also propose some new weightedaveraging methods for gradient recovery,and present analytical and numerical investigation in theperformance oftheseweighted averagingmethods．Itis shownthat,analytically,theareaharmonicaveraging yields a superconvergent gradient for any mesh in one-dimension and the rectangularmesh in two-dimension．Numerical study indicates that these new weighted averaging methodsresult in better recovered gradient than simple averaging and area averaging under the triangular mesh．We propose a novel approach to recover normal derivatives for a smooth function based on itspiecewise L~2 projection．For each polynomial(of degree up to 4)projection,another polynomialof same degree(at least degree one)is constructed over a sub-domain centered at the interfaceseparating two polynomials,its normal derivative at interface is taken to be the recovered normalderivative．From such a recovery algorithm we obtain a set of numerical flux fornmlae for solutionderivatives．We apply these flux fommlae to the direct discontinuous Galerkin(DDG)methods forsome elliptic problems using polynomial elements of degree up to 4．Some adaptation of thesenumerical flux is adopted for even high order elements．Both one and two-dimensional numericalresults are provided to demonstrate the good qualities of the recovery algorithm when combinedwith the DDG methods．We also applied the new gradient recovery techniques to a posteriori error estimation andadaptive finite element methods．We use the adaptive methods to some elliptic equation．It isshown that the new recovery type error estimators are efficient,refiable and asymptotically exact．We study the adaptive methods for elliptic control problems．For the control problem wim allintegral constraint,we derive a priori estimates and a posteriori error estimates for the spectral ap-proximation of the optimal control problems．And we consider the adaptive finite element methodfor the control problems with pdintwise constraint,two recovery type a posteriori error estima-tors are proposed．Numerical examples are presented for illustrating the efficiency of the adaptivealgorithm．...