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Vanishing moment method and moment solutions for second order fully nonlinear partial differential equations
Xiaobing Feng, Michael Neilan
TL;DR
Numerical approximation of general second-order fully nonlinear PDEs remains difficult, in part because viscosity solutions are not constructive or variational. The paper introduces moment solutions through a vanishing moment method that approximates these equations by higher-order quasilinear PDEs. Its numerical experiments report convergence and agreement with viscosity solutions whenever those solutions exist.
Problem
Numerically approximating general second-order fully nonlinear PDEs is difficult because viscosity solutions are not constructive or variational, leaving no direct computational handle.
Source: §1 · p. 2, §2.1 · p. 5Method
The vanishing moment method defines moment solutions as limits of solutions to higher-order, particularly fourth-order, quasilinear PDE approximations.
Source: §1 · p. 3Results
Numerical experiments report convergence of the vanishing moment method and coincidence of moment and viscosity solutions whenever the latter exist.
Source: §5.1 · pp. 13, 15, §5.2 · p. 21Takeaways & Limitations
The constructive limiting process provides a weak-solution concept for second-order fully nonlinear PDEs that can be computed through higher-order PDE approximations.
Source: §1 · p. 3Numerical methods and experiments are restricted to spatial dimensions n = 2 and 3.
Source: §2.1 · p. 4
Abstract
from arXiv · showhide
This paper concerns with numerical approximations of solutions of second order fully nonlinear partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for second order fully nonlinear PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called vanishing moment method, hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods with "guaranteed" convergence. The main idea of the proposed vanishing moment method is to approximate a second order fully nonlinear PDE by a higher order, in particular, a fourth order quasilinear PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.
1. Introduction.
Although viscosity-solution theory has advanced for second-order fully nonlinear PDEs, their numerical approximation remains difficult because viscosity solutions lack a discrete-level equivalent and standard weak formulations are unavailable. The paper introduces constructive moment solutions through the vanishing moment method and presents numerical methods and experiments to study their convergence and agreement with viscosity solutions.
1. Introduction.
- Numerical approximation remains difficult because direct discretization can lose control of the targeted solution, while viscosity solutions lack a discrete equivalent and Galerkin methods lack a variational formulation.Strong nonlinearity and nonuniqueness further complicate the resulting algebraic problems.Source: §1 · pp. 2–3
- Earlier approaches include specialized finite-difference schemes and variational methods, but convergence analysis was generally unaddressed; Bellman methods rely on operator linearity and do not extend to other equation types.The cited convergence analysis among these approaches was addressed in.Source: §1 · p. 3
- The vanishing moment method constructs moment solutions by approximating a second-order fully nonlinear PDE with higher-order, particularly fourth-order, quasilinear PDEs.A moment solution is the limit of the higher-order PDE solution sequence, when that limit exists.Source: §1 · p. 3
- The paper presents numerical methods and experiments to demonstrate convergence and effectiveness, and studies whether moment solutions coincide with viscosity solutions whenever the latter exist.The method is intended to let existing numerical methods and codes for fourth-order linear and quasilinear PDEs be used.Source: §1 · p. 4
2. Vanishing moment method and the notion of moment solutions.
The section contrasts the nonconstructive viscosity-solution framework with a constructive limit-based approach for second-order fully nonlinear PDEs. It defines moment solutions through higher-order quasilinear approximations and discusses choices and unresolved questions in that construction.
2.1. Preliminaries.
- Viscosity solutions are difficult to approximate directly for general second-order fully nonlinear PDEs because they are neither constructive nor variational.The modern definition is based on “differentiation by parts,” rather than the traditional integration-by-parts formulation.Source: §2.1 · p. 5
2.2. General framework of the vanishing moment method.
- The vanishing moment method approximates a second-order fully nonlinear PDE by higher-order quasilinear PDEs, with the added operator tending to zero as ε tends to zero.The framework calls for a linear or quasilinear operator Gε; the paper uses a fourth-order formulation.Source: §2.2 · p. 6
- A limit of the approximating solutions, when it exists, is a moment solution; L2, H1, and H2 convergence define sub-weak, weak, and strong moment solutions, respectively.Strong moment solutions have second-order weak derivatives and are expected to satisfy the PDE almost everywhere and the boundary condition pointwise.Source: §2.2 · pp. 6–7
- The framework leaves operator and boundary-condition choices, limit existence, and convergence rates as open issues; this paper focuses on developing the framework and presenting numerical evidence.The authors state that they do not address all theoretical questions here and plan to treat them in forthcoming papers.Source: §2.2 · p. 6
- The fourth-order Dirichlet approximation needs an additional boundary condition, and the proposed choices aim to minimize the boundary layer while perturbing the original Dirichlet condition by higher-order terms.The paper notes that an essential Neumann condition produces a larger boundary layer than the other proposed choices.Source: §2.2 · p. 7
3. Discretization and solution methods.
The paper discretizes the fourth-order nonlinear biharmonic problems using existing Galerkin methods, including conforming and mixed finite elements. It also extends these discretizations to parabolic equations by time stepping, while noting challenges for fully discrete convergence and error analysis.
3. Discretization and solution methods.
- For the fourth-order nonlinear biharmonic equation, the paper focuses on Galerkin discretizations, including finite element, mixed finite element, discontinuous Galerkin, and spectral Galerkin methods.The discussion assumes n = 2, 3.Source: §3 · p. 8, §3.2 · p. 9
3.1. Finite element methods in 2-d.
- In two dimensions, the conforming finite element method adapts established plate elements to the vanishing-moment problem, with numerical experiments planned for Monge-Ampère type equations.Convergence and error analysis are deferred to forthcoming work, also cited as.Source: §3 · p. 8, §3.1 · p. 8
3.2. Mixed finite element methods in 2-d and 3-d.
- Because the Hessian enters the equation nonlinearly, the mixed method introduces σε := D2uε as an additional variable and thereby excludes Ciarlet-Raviart elements.The paper discusses Hermann-Miyoshi and Hermann-Johnson mixed elements using this additional variable.Source: §3.2 · pp. 8–9
3.3. Remarks on second order fully nonlinear parabolic equations.
- For parabolic equations, implicit time stepping reduces each time step to solving a fully nonlinear elliptic equation, so the preceding numerical methods apply.Fully discrete convergence and error analysis are expected to be harder, especially polynomial rather than exponential dependence of error estimates on ε−1.Source: §3.2 · p. 9
3. Discretization and solution methods.
- The numerical experiments use preconditioned Newton iterations with ILU preconditioning; the authors are investigating a multiresolution initialization strategy and plan to explore stronger preconditioners for larger 3-D computations.A few fixed-point iterations may be used to generate Newton initial guesses.Source: §3.3 · p. 10
4. Applications.
The vanishing moment method is applied to Monge–Ampère-type equations, Pucci’s equations, the infinite Laplace equation, and a class of fully nonlinear parabolic PDEs. The supplied results establish agreement with viscosity solutions for the classical Monge–Ampère equation in two dimensions and for the infinite Laplace equation; the parabolic cases are formulated as fourth-order quasilinear approximations.
4.1. Monge-Amp`ere type equations.
- In two dimensions, the classical Monge–Ampère problem has a unique convex strong moment solution, coinciding with its unique convex viscosity solution.For each fixed ε > 0, the approximation has a unique weak solution in W 3,2(Ω) for n = 2, 3; the stated convexity and determinant properties also hold for the indicated dimensions.Source: §4.1 · pp. 10–11
4.2. Pucci’s equations.
- The vanishing moment approximations for Pucci’s extremal equations are paired with the specified boundary conditions.The passages introduce approximations for both equations but do not report their numerical outcomes here.Source: §4.2 · p. 11
4.3. Infinite Laplace equation.
- For the infinite Laplace equation, the vanishing moment approximation converges exactly to the unique viscosity solution of its Dirichlet problem.The passage presents this as another case where moment and viscosity solutions coincide.Source: §4.3 · p. 12
4.4. Second order fully nonlinear parabolic PDEs.
- For the considered fully nonlinear parabolic equations, the proposed vanishing moment approximations are fourth-order quasilinear parabolic PDEs.The paper notes that numerical approximation of these equations had not previously been attempted, to the authors’ knowledge.Source: §4.4 · p. 12
5. Numerical experiments.
The experiments apply the vanishing moment method to two- and three-dimensional fully nonlinear PDEs using Argyris and Hermann-Miyoshi finite elements. Across several tests, computed moment solutions match viscosity solutions when those exist; tests also examine degenerate and type-changing problems.
5.1. Two-dimensional numerical experiments.
- In two-dimensional tests, vanishing moment approximations capture viscosity solutions, with moment and viscosity solutions coinciding in the reported cases.The tests include convex and concave solutions and an infinite Laplace example.Source: §5.1 · pp. 13, 14, 18
- For a degenerate Monge–Ampère problem, the approximations capture convex and concave viscosity solutions, and the method is reported robust to degeneracy.The forcing vanishes along x + y = 1.Source: §5.1 · p. 15
- For a type-changing Monge–Ampère problem, numerical results indicate both convex and concave moment solutions, although the authors report no viscosity solution theory for this problem type.The forcing changes sign and vanishes along x − y = 0.Source: §5.1 · pp. 15–16
- For the Gauss-curvature test, computed convex solutions are reported for K = 0.1, 1, 2, and 2.1; the code fails to produce a convergent numerical solution at K = 2.2.The authors conjecture K∗ ≈ 2.1 for this test problem.Source: §5.1 · p. 17
- The quadratic Hermann-Miyoshi mixed method gives the same accuracy as Argyris and runs about 20 times faster on the reported tests.It also approximates the Hessian matrix, which is not shown.Source: §5.1 · pp. 18–19
5.2. Three-dimensional numerical experiments.
- In both three-dimensional tests, generalized Hermann-Miyoshi mixed methods produce moment solutions reported to coincide with the convex viscosity solutions.One test has a unique exact convex solution; the other has no explicit solution formula and no classical solution.Source: §5.2 · pp. 20–21
6. Conclusions.
The paper introduces moment solutions for second order fully nonlinear PDEs through the constructive vanishing moment method, which approximates them with higher order quasilinear PDEs. It presents numerical methods and experiments to assess convergence and the relationship between moment and viscosity solutions.
6. Conclusions.
- The vanishing moment method approximates second order fully nonlinear PDEs with higher order quasilinear equations and defines moment solutions through this constructive limit.The paper applies its framework to Monge-Ampère type equations, Pucci’s extremal equations, the infinite Laplace equation, and second order fully nonlinear parabolic PDEs.Source: §6 · p. 21
- The paper presents a practical, systematic methodology for approximating second order fully nonlinear PDEs, backed by rigorous PDE and numerical theories.Source: §6 · p. 21
- Moment solution theory may inform viscosity solution theory and extend the notion of viscosity solutions to cases where theories are absent or fail, including certain Monge-Ampère equations and fully nonlinear systems.The paper names subelliptic and hyperbolic Monge-Ampère equations as examples.Source: §6 · p. 22