| The fourth order elliptic equations frequently occur in the fields of solid mechanics, ma-terial science, image processing and so on. It is both theoretically and practically importantto investigate numerical methods for such equations. This thesis is intended to construct anefficient Poisson based solver for the biharmonic equation discretized by Morley's element,to design local and parallel algorithms for fourth order problems discretized by the MWXelement method, to present general C0 discontinuous Galerkin method for plate bendingproblems, and to give a two-level additive Schwarz preconditioner for local C0 discontinu-ous Galerkin method of plate bending problems. Also, a general framework is establishedfor adaptive finite element methods, which can cover almost all existing typical AFEMs.Firstly, an efficient Poisson based solver is presented for the biharmonic equation dis-cretized by Morley's element. By the minimum principle of total potential energy, an equiva-lence relation is established between the biharmonic equation and the Stokes equation. Afterthat, a similar equivalence analogue is obtained between the numerical method for the bihar-monic equation discretized by Morley's element and that of the Stokes equation discretizedby the nonconforming P1 ? P0 element. Then, by use of this equivalence together withsome algebraic multigrid methods, an efficient Poisson based solver is given for solving thebiharmonic equation discretized by Morley's element. The efficiency of the new solver isillustrated by some numerical experiments.Secondly, three two-grid local and parallel algorithms are designed for solving fourthorder problems discretized by the MWX element method in any dimensions. Since the MWXelement spaces are nonnested, some intergrid transfer operators have to be introduced, in or-der to get an improved global solution from the global coarse grid solution and the local finegrid corrections. Concretely speaking, three two-grid local and parallel algorithms are pro-posed for solving fourth order problems discretized by the MWX element method, throughintroducing a modified Argyris element based intergrid transfer operator. It is shown that thediscrete energy error of the numerical solution is bounded by O(h + H2). Then, based oncertain arithmetic average intergrid transfer operators, three two-grid local and parallel algo-rithms are proposed, and the discrete energy error of the numerical solution is bounded by O(h + H2(H/h)(d-1)/2). Furthermore, a number of numerical results are reported to showthe computational performance of the methods just mentioned.Thirdly, we develop the general C0 discontinuous Galerkin method for plate bendingproblems and give the error estimate for the numerical solution in broken energy norm andH1 norm. We first reformulate the original fourth-order partial differential equation as asecond-order system and obtain a framework of constructing CDG methods for solving theoriginal problem. Then, we establish a discrete stability identity, from which we derive fea-sible choices of numerical traces and get a class of stable CDG methods for plate bendingproblems. Following some ideas on error analysis of DG methods for second order ellipticproblems and detailed technical derivation, we derive error estimates of the numerical solu-tion in certain broken energy norm and H1 norm for the LCDG method and CDG method.Some numerical results are included to confirm our theoretical convergence orders.Moreover, a two-level additive Schwarz preconditioner is constructed for the above-mentioned local C0 discontinuous Galerkin method of plate bending problems. By con-structing a special intergrid transfer operator, a domain decomposition method is developedfor plate bending problems discretized by our LCDG method. And we estimate the conditionnumber of the coefficient matrix of the preconditioned linear system. The upper bound ofthe condition number will be optimal whenδ≈H. In the case of small overlap, i.e.δ<< H,the upper bound of the condition number can be improved in view of some more carefulmathematical reasoning.Then, a general framework is developed for adaptive finite element methods. Undersome assumptions, we get the qausi-optimal convergence rate of adaptive finite elementmethods. With some more assumptions, by introducing the total error, the optimal com-plexity of adaptive finite element methods is achieved too. At last, the general framework isapplied to several partial differential equations, including general second order elliptic equa-tions with high order finite elements, 2m-th order elliptic problems with Morley-type finiteelements, classical time-harmonic Maxwell's equations, and H(div) equations.Finally, the assumptions in the general framework of adaptive finite element methodsare verified for adaptive mixed finite element method with k = 0, 1 for plate bending prob-lems. We make use of the Helmholtz decomposition to construct a residual-based a posteriorierror estimator for a mixed finite element method, namely the H-H-J method with all ordersk≥0, and prove its reliability. Then the efficiency of the error estimator is achieved bythe technique of the bubble functions. Furthermore, an improved version is provided for thesubsequent adaptive algorithm based on the property that the force jump across an interior edge is dominated by the volume part. Moreover, a discrete Helmholtz decomposition and adiscrete inf- sup condition are derived, and then by means of these two tools and the prop-erty obtained in the a posteriori analysis, we are able to establish the quasi-orthogonality andeven the discrete reliability. Then, we develop the quasi-optimal convergence theory as wellas the optimal complexity for the H-H-J method.
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