Xia, Shiqiang2022-08-292022-08-292022-04https://hdl.handle.net/11299/241349University of Minnesota Ph.D. dissertation. May 2022. Major: Mathematics. Advisor: Bernardo Cockburn. 1 computer file (PDF); x, 137 pages.In this dissertation, we study an adjoint-based method for the computation of highly accurate approximations of functionals of solutions of second-order elliptic problems. First, we present a rigorous a priori error analysis of the method applied to linear functionals. We also carry out comprehensive numerical experiments to display the super-convergence properties of the method and to demonstrate the advantage of the method over the standard method. We then extend the method to an important nonlinear problem, namely, that of the computation of eigenvalues. To handle the possible lack of smoothness of the solution or that of the functional, we further develop a new mesh adaptative algorithm which we incorporate into the adjoint-based method.enAdjoint-based methodConvolutionDiscontinuous Galerkin methodsGoal-oriented error approximationOutput functionalssuperconvergentThesis or Dissertation