hp-version interior penalty DGFEMs for the biharmonic equation
ABSTRACT We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments.
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ABSTRACT: A general framework of constructing C0 discontinuous Galerkin (CDG) methods is developed for solving the Kirchhoff plate bending problem, following some ideas in (Castillo et al., 2000)  and (Cockburn, 2003) . The numerical traces are determined based on a discrete stability identity, which lead to a class of stable CDG methods. A stable CDG method, called the LCDG method, is particularly interesting in our study. It can be viewed as an extension to fourth-order problems of the LDG method studied in (Castillo et al., 2000)  and (Cockburn, 2003) . For this method, optimal order error estimates in certain broken energy norm and H1-norm are established. Some numerical results are reported, confirming the theoretical convergence orders.Computer Methods in Applied Mechanics and Engineering.
Article: Discontinuous Galerkin Finite Element Approximation of the Cahn-Hilliard Equation with Convection.SIAM J. Numerical Analysis. 01/2009; 47:2660-2685.
J. Sci. Comput. 01/2008; 37:139-161.