**Contact Info**

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Phone: 519-888-4567, ext. 32700

Fax: 519-746-4319

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Tuesday, September 1, 2015 — 1:00 PM EDT

MC 6496

Noel Chalmers | Applied Math, University of Waterloo

Superconvergence, Superaccuracy, and Stability of the Discontinuous Galerkin Finite Element Method

This thesis is concerned with the investigation of the superconvergence, superaccuracy, and stability properties of the discontinuous Galerkin (DG) finite element method in one and two dimensions. We propose a novel method for the analysis of these properties. We apply the DG method to a model linear advection problem to derive a PDE which is satisfied by the numerical solution itself. This PDE is equivalent to the original advection equation but with a forcing term that is proportional to the jump in the numerical solution at the cell interfaces. We then use classical Fourier analysis to determine the solutions to this PDE with particular temporal frequencies. We find that these Fourier modes are completely determined on each cell by the inflow into that cell and a certain rational function of the mode's frequency. By using local expansions of these modes, we prove several local superconvergence properties of the DG method, as well as superaccurate errors in terms of dissipation and dispersion. Next, by considering a uniform mesh and assuming periodic boundary conditions, we investigate the spectrum of the method. In particular, we show that the spectrum can be partitioned into physical and non-physical modes. The physical modes advect with high-order accuracy while the non-physical modes decay exponentially quickly in time. Using these results we establish several global superconvergence properties of the method on uniform meshes. Finally, we also propose a new family of schemes which can been viewed as a modified version of the DG scheme. We extend our analysis to these new schemes we construct schemes with significantly larger stable CFL numbers than the classic DG method. We demonstrate through some numerical examples that these modified schemes can be effective in capturing fine structures of the numerical solution when compared with the DG scheme with equivalent computational effort.

**Contact Info**

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Phone: 519-888-4567, ext. 32700

Fax: 519-746-4319

PDF files require Adobe Acrobat Reader

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

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