Nonstandard local discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions

TL;DR

This paper develops high-order local discontinuous Galerkin methods for solving fully nonlinear second order PDEs in multiple dimensions, addressing low regularity of solutions with multiple derivative approximations.

Contribution

It introduces a novel LDG framework with consistency and monotonicity properties for fully nonlinear PDEs, extending narrow-stencil finite difference approaches.

Findings

Methods accurately approximate viscosity solutions.

Numerical tests demonstrate high accuracy and efficiency.

The algebraic system structure facilitates nonlinear solver design.

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.