Numerical Analysis of Discontinuous Galerkin Methods for Second Order Linear Non-divergence Form PDEs

Abstract

Non-divergence form PDEs arrive in plenty of applications such as stochastic optimal control and mathematical finance. They also appear in the study of fully non-linear partial differential equations associated with linearization techniques. However, compared with the advances of the PDE analysis in divergence form, there is less progress on numerical methods and analysis for second order elliptic and parabolic PDEs in non-divergence form with non-smooth coefficient matrix. The non-divergence structure of the PDEs prevents some straightforward applications of Galerkin-type numerical methodologies. More precisely, finite element methods, discontinuous Galerkin methods and spectral methods are not easy to implement for non-divergence form PDEs with non-differential coefficient matrices. Moreover, the concepts of the strong and viscosity solutions are non-variational, which is also difficult for us to do convergence analysis and error estimates of any convergent numerical method.

In this thesis, we provide numerical analyses of the proposed discrete narrow-stencil scheme for the underlying second order elliptic and parabolic non-divergence form equations, respectively.

For the second order linear elliptic partial differential equations in non-divergence form with continuous coefficient matrix, we analyze a narrow-stencil discontinuous Galerkin method to approximate W^2,p strong solutions (for some 1<p<∞). The main novelty of our analyses is to construct the relationship between two types of mesh-dependent norms: one is the standard discrete W^1,p and W^2,p norms; the other is associated with the narrow-stencil DG method. This novelty is one crux of our whole analysis since we can't directly utilize conventional discrete inverse and trace inequalities for the discrete derivatives of DG functions defined in the proposed narrow-stencil DG scheme. By combining this important relationship, some basic properties of the DG space and a freezing coefficient technique, a DG discrete counterpart of the Calderon-Zygmund estimate is established under the assumption that the coefficient matrix is globally continuous and the Calderon-Zygmund estimate is satisfied. Then it is proved that the proposed narrow-stencil DG methods converge with optimal rate to the W^2,p strong solution in the standard discrete W^2,p-norm for some 1<p<∞.

For the second order linear non-divergence form parabolic PDEs with continuous coefficient matrix, we approximate the viscosity solutions by a narrow-stencil finite difference spatial-discretization paired with the forward Euler method for the time discretization. By utilizing generalized monotonicity of the numerical scheme and an iteration technique, we prove the numerical scheme is stable in both the l²-norm and the l^∞-norm. Then under this stability, we establish the convergence of the proposed numerical scheme to the viscosity solution of the underlying second order parabolic non-divergence form PDEs based on different regularities with respect to time and space. We also report some numerical experiments.

Date of Award	7 May 2024
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Hui-Hui DAI (Supervisor) & Weifeng QIU (Supervisor)