Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations

TL;DR

This paper develops high-order local discontinuous Galerkin methods for accurately solving fully nonlinear second order elliptic and parabolic PDEs in one dimension, capturing derivative discontinuities and ensuring stability.

Contribution

It introduces a general LDG framework with a novel mixed formulation and criteria for constructing stable numerical operators for fully nonlinear PDEs.

Findings

Framework extends to high order polynomials and non-uniform meshes.

Ensures stability via a generalized monotonicity criterion.

Captures derivative discontinuities effectively.

Abstract

This paper is concerned with developing accurate and efficient discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in the case of one spatial dimension. The primary goal of the paper to develop a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs which are merely continuous functions by definition. In order to capture discontinuities of the first order derivative $u_x$ of the solution $u$, two independent functions $q_1$ and $q_2$ are introduced to approximate one-sided derivatives of $u$. Similarly, to capture the discontinuities of the second order derivative $u_{xx}$, four independent functions $p_{1}$, $p_{2}$, $p_{3}$, and $p_{4}$ are used to approximate one-sided derivatives of $q_1$ and $q_2$. The proposed LDG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a given fully nonlinear problem into a mostly linear system of equations where the given nonlinear differential operator must be replaced by a numerical operator which allows multiple value inputs of the first and second order derivatives $u_x$ and $u_{xx}$. An easy to verify criterion for constructing "good" numerical operators is also proposed. It consists of a consistency and a generalized monotonicity. To ensure such a generalized monotonicity, the crux of the construction is to introduce the numerical moment in the numerical operator. The proposed framework extends a companion finite difference framework developed by the authors in [9] and allows for the approximation of fully nonlinear PDEs using high order polynomials and non-uniform meshes.