Goal-oriented Adaptive Finite Element Methods For Elliptic Equations

Goal-oriented adaptive finite element method(GOAFEM)is an adaptive finite element method(AFEM)specially designed for estimating errors of numerical approximations of partial differential equations in goal functionals.This approach is used in many scientific and engineering applications.In this dissertation,we study GOAFEMs for second-order linear and semilinear elliptic equations.For linear elliptic problems,we are interested in the value of the solution at the given point or the integral of the solution along the given line in the domain.For the point quantity of interest,we propose and numerically validate two AFEMs based on the residual-type and recovery-type goal-oriented a posteriori error estimators,respectively.As for the line quantity of interest,we also propose and numerically validate an AFEM with a reliable goal-oriented a posteriori error estimator.The highlight of adaptive algorithms for point and line quantities of interest,is using a dual problem with the point and line Dirac delta source term,to estimate the corresponding error.Especially,in contrast to GOAFEMs based on the classical mollification technique,those two pointwise GOAFEMs assess the point-value error more accurately.This advantage is shown with numerical comparisons.For semilinear elliptic problems,the quantities of interest under consideration are the linear or nonlinear integrals of the solution in the given subdomain.For the linear quantity of interest,following the adaptive two-grid idea which regards the kth and the(k + 1)th adaptive meshes as the coarse and fine meshes,respectively,we propose and numerically validate adaptive two-grid finite element methods(ATGFEMs)based on the one-step correction of Newton iteration,Zarantonello iteration and Ka(?)anov iteration.On each adaptive mesh except the initial mesh,of all three adaptive algorithms,the discrete primal problem is linear,and the discrete dual problem based on the linearization related to nonlinear iterations has the same coefficient matrix as the corresponding discrete primal problem.The proposed GOATGFEMs are more efficient than classical GOAFEMs with enough nonlinear iterations.We establish the convergence theory of the GOATGFEM with the one-step Newton correction.For the nonlinear quantity of interest,we propose two h–h/2 type AFEMs,which introduce a globally refined auxiliary mesh for each current adaptive mesh.We construct goal-oriented a posteriori error estimators by means of replacing the unkown exact primal and dual solutions in the error representation of the nonlinear quantity of interest by their finite element approximations on the auxiliary fine mesh instead of using the conventional estimator product or DWR method.We prove the error estimator of the first algorithm to be the lower and upper bounds of the error in the quantity of interest.The error estimator of the second algorithm is more efficient computationally than the one of the first algorithm.Numerical experiments show the effectiveness and the similar convergence performance of the proposed algorithms.