Multiscale discontinuous Galerkin methods and applications Wang, Wei creator 2008 xi, 91 p. born digital Thesis (Ph.D.) -- Brown University (2008) Shu, Chi-Wang director Gottlieb, David reader Hesthaven, Jan reader Brown University. Applied Mathematics sponsor theses This thesis contains three related topics on the multiscale discontinuous Galerkin (DG) methods and applications. In the first part, we present a multiscale model for numerical simulation of dynamics of crystalline solids. The method couples nonlinear elastodynamics as the continuum description and molecular dynamics as another component at the atomic scale. The governing equations on the macroscale are solved by the DG method, which is built up with an appropriate local curl-free space to produce a coherent displacement field. In the second part, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the 1-D stationary Schr\"{o}dinger-Poisson problem. The WKB-LDG method is proposed for solving the Schr\"{o}dinger equation and provides a significant reduction of both the computational cost and memory. It has the advantages of the DG methods including their flexibility in $h$-$p$ adaptivity and the allowance of complete discontinuity at element interfaces compared with a traditional continuous finite element Galerkin methodology. In the third part, we develop a multiscale DG method for solving a class of second order elliptic problems with rough coefficients based on the previous work of Yuan and Shu \cite{YS2}. The main ingredient of this method is to use a non-polynomial multiscale approximation space in the DG method to capture the solutions without resolving the fine-scale structure of the solution. We generalize the analysis of the multiscale Babu\v{s}ka-Zl\'amal DG method to the case of $u\in H^1([0,1])$. We also propose a multiscale local discontinuous Petrov-Galerkin method and a multiscale interior penalty DG method in the numerical tests. multiscale discontinuous Galerkin methods RPB 20091218 engEnglish10.7301/Z0PN93XKIn CopyrightCollection is open for research.dissertations