Mixed Interior Penalty Discontinuous Galerkin Methods for One-Dimensional Fully Nonlinear Second Order Elliptic and Parabolic Equations

Abstract

Abstract. This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for constructing high order interior penalty discontinuous Galerkin (IP-DG) methods for approximating viscosity solutions of these fully nonlinear PDEs. In order to capture discontinuities of the second order derivative uxx of the solution u, three independent functions p 1 , p 2 and p 3 are introduced to represent numerical derivatives using various one-sided limits. The proposed DG framework, which is based on a nonstandard mixed formulation of the underlying PDE, embeds a nonlinear problem into a mostly linear system of equations where the nonlinearity has been modified to include multiple values of the second order derivative uxx. 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. In addition to the nonstandard mixed formulation setting, another main idea is to replace the fully nonlinear differential operator by a numerical operator, which is consistent with the differential operator and satisfies certain monotonicity (called g-monotonicity) properties. To ensure such a g-monotonicity, the crux of the construction is to introduce the numerical moment, which plays a critical role in the proposed DG framework. The g-monotonicity gives the DG methods the ability to select the mathematically "correct" solution (i.e., the viscosity solution) among all possible solutions. Moreover, the g-monotonicity allows for the possible development of more efficient nonlinear solvers as the special nonlinearity of the algebraic systems can be explored to decouple the equations. This paper also presents and analyzes numerical results for several numerical test problems which are used to guage the accuracy and efficiency of the proposed DG methods.Key words. Fully nonlinear PDEs, viscosity solutions, discontinuous Galerkin methods, AMS subject classifications. 65N30, 65M60, 35J60, 35K55, 1. Introduction. Fully nonlinear partial differential equations (PDEs) refer to a class nonlinear PDEs which is nonlinear in the highest order derivatives of the unknown functions in the equations. Due to their strong nonlinearity, this class of PDEs are most difficult to analyze analytically and to approximate numerically. In the mean time, fully nonlinear PDEs arise in many applications such as antenna design, astrophysics, differential geometry, fluid mechanics, image processing, meteorology, mesh generation, optimal control, optimal mass transport, etc [8], which calls for the development of efficient and reliable numerical methods for solving their underlying fully nonlinear PDE problems. This is the second paper in a series [9] which is devoted to developing finite difference (FD) and discontinuous Galerkin (DG) methods for approximating ...

Mixed interior penalty discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions

“…Since the numerical moment is the key to designing a consistent, gmonotone numerical operator, we will also use the conditionally elliptic Monge-Ampère test problems to better understand the contribution of the numerical moment. The tests in the 1D case, see [3], indicate the spatial error may have order , where However, the rates are not perfectly clear from the test data in this article. The results for approximating the problem for r = 1, r = 2, and r = 3 are recorded in Table 1.…”

“…The discretization techniques for fully nonlinear PDEs and the choice of solver for the resulting nonlinear systems of equations should not be considered entirely independent. We see in many tests that the addition of a numerical moment yields a system of equations that is better suited for generic Newton solvers, especially when considering the 1D numerical tests in [3]. However, the tests in Section VI further indicate that the numerical moment has a much greater impact for approximating fully nonlinear PDEs when used in concert with an appropriate solver.…”

“…Using a numerical operator that can handle multiple approximations for , we rewrite (3.1) in mixed form as We now formalize the definition and properties of a numerical operator. The following definitions are high-dimensional counterparts of the one-dimensional (1D) definitions in [3]. a.…”

“…Then, we have Ph and correspond to two IP-DG approximations for the Hessian of u that both use averaged flux values on all interior faces/edges; i.e., Ph and are both solutions to (3.10) using the empty value for *, where the only difference is the value of the penalty parameter γ. An essential theme in the series [2][3][4] devoted to approximating fully nonlinear second-order PDEs is the ability to form at least two different Hessian approximations.…”

“…Numerical tests for the proposed IP-DG methods for parabolic PDEs of type (1.2) can be found in [3]. In all of our tests, we use uniform Cartesian partitions on rectangular domains.…”