Nonstandard Local Discontinuous Galerkin Methods for Fully Nonlinear Second Order Elliptic and Parabolic Equations in High Dimensions

Abstract

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

A narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

“…We then state a few structure conditions/assumptions for numerical operators. It should be noted that all of these concepts and conditions are motivated by and abstractions of similar ones in our earlier works [11,12,14,24]. The numerical moment will play a critical role in the specific examples of numerical operators that are presented.…”

“…We note that the above FD numerical derivative operators are only defined on uniform Cartesian grids. In order to extend them to arbitrary grids, and, in particular, to triangular/tetrahedral grids, we utilize finite element DG numerical derivatives which were first introduced in [16] (also see [12]). Below we recall their definitions and some useful properties, but we shall use new notations which are consistent with the above FD discrete derivative operators.…”

“…We call the second approach the numerical moment-enhanced g-monotone (narrow-stencil) finite difference and DG framework (cf. [11,12,14,24] and also the original vanishing moment method [17]). It is well-known that monotone (in the sense of Barles-Souganidis [1]) methods are difficult to construct; moreover, they are intrinsically low order, require the use of wide stencils, and yield strongly coupled nonlinear algebraic problems that need to be solved.…”

“…On the other hand, since the narrow-stencil approach abandons the standard monotonicity requirement, it prevents one to directly use the powerful Barles-Souganidis' framework for the convergence analysis. Consequently, new machineries and techniques must be developed for the analysis of the proposed narrow-stencil methods which, so far, has only been done on a case-by-case basis in [11,12,14,24]. The primary goal of this paper is to re-examine (with a top-down view) and refine the numerical moment-enhanced g-monotone (narrow-stencil) finite difference and DG approach, which was initiated by us in [11,12,14,24], and to formulate it into a unified framework which is parallel to the Barles-Souganidis' framework.…”

“…Consequently, new machineries and techniques must be developed for the analysis of the proposed narrow-stencil methods which, so far, has only been done on a case-by-case basis in [11,12,14,24]. The primary goal of this paper is to re-examine (with a top-down view) and refine the numerical moment-enhanced g-monotone (narrow-stencil) finite difference and DG approach, which was initiated by us in [11,12,14,24], and to formulate it into a unified framework which is parallel to the Barles-Souganidis' framework. A far-reaching goal is to provide a blueprint/framework for designing and analyzing practical convergent numerical methods for approximating viscosity (and regular) solutions of fully nonlinear second order PDEs.…”