(January 31) Dr. Weifeng (Frederick) Qiu: An analysis of the practical DPG method — In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a ``trial-to-test'' operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r> p+N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.

(February 7) Wujun Zhang: Convergence and quasi optimality of adaptive hybridizable discontinuous Galerkin methods — We establish the convergence and quasi optimality of adaptive hybridizable discontinuous Galerkin (AHDG) methods for the Poisson problem. We prove that the so-called quasi-error, that is, the sum of an energy-like error and a scaled error estimator, is a contraction between two consecutive loops. Moreover, we show that the AHDG methods achieve optimal rates of convergence.

(February 14) Prof. Soeren Bartels: Finite element approximation of large bending isometries — The mathematical description of the elastic deformation of thin plates can be derived by a dimension reduction from three-dimensional elasticity and leads to the minimization of an energy functional that involves the second fundamental form of the deformation and is subject to the constraint that the deformation is an isometry. We discuss two approaches to the discretization of the second order derivatives and the treatment of the isometry constraint. The first one relaxes the second order derivatives via a Reissner-Mindlin approximation and the second one employs discrete Kirchhoff triangles that define a nonconforming second order derivative. In both cases the deformation is decoupled from the deformation gradient and this enables us to employ techniques developed for the approximation of harmonic maps to impose the constraint on the deformation gradient at the nodes of a triangulation. The solution of the nonlinear discrete schemes is done by appropriate gradient flows and we demonstrate their energy decreasing behaviours under mild conditions on step sizes. Numerical experiments show that the proposed schemes provide accurate approximations for large vertical loads as well as compressive boundary conditions.

(March 13) Prof. Dianne O'Leary: TBA — TBA

webmaster@math || Math Department || Numerical Analysis || Seminars