Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations

A convergent finite difference method for computing minimal Lagrangian graphs

“…For this reason, we choose to utilize piecewise Cartesian meshes augmented with additional nodes along the boundary. These can be conveniently stored using a quadtree structure as in [19]. See Figure 2 for examples of such meshes.…”

“…In order to construct consistent numerical methods, we require that h B = o(h) and δ = O(h) as h → 0. This is easily accomplished as described in [19]. We will also associate to the mesh a directional resolution dθ and a search radius r, whose roles will become clear in the remainder of this section.…”

“…From the same section in [4], we know that the maximum in (1. 19) is attained when n = n y . Since we know that n x • n y ≥ > 0, we can restrict the maximum to vectors n satisfying this constraint.…”

“…In particular, we use this framework to introduce and implement a numerical method for computing minimal Lagrangian submanifolds. The method utilizes generalized finite difference approximations on augmented piecewise-Cartesian grids, as introduced in [19]. We adapt this to the minimal Lagrangian problem, and show how this approach can be used to enforce the second boundary condition (0.4) and numerically compute the eigenvalue c. Though the method is implemented in two dimensions, it could be easily adapted to higher dimensions and more complicated PDEs.…”

A convexity enforcing $${C}^{{0}}$$ interior penalty method for the Monge–Ampère equation on convex polygonal domains

“…There are many numerical approaches to the Dirichlet boundary value problem of the Monge-Ampère equation (and related equations) in 2 and 3 spatial dimensions, with respect to different solution classes (classical solutions, Aleksandrov solutions [2] and viscosity solutions [54]). They include (i) geometric finite difference methods [63,66,68,69], (ii) monotone finite difference methods [7][8][9][39][40][41]48,50,67], (iii) augmented Lagrangian and least-squares finite element methods [19,[28][29][30][31], (iv) finite element methods based on the vanishing moment approach [3,[35][36][37]57], (v) finite element methods based on L 2 projection [4,5,10,11,13,15,27,51,[58][59][60], (vi) finite element methods based on a reformulation of the Monge-Ampère equation as a Hamilton-Jacobi-Bellman equation [14,34], and (v) two-scale methods [53,64,65]. Comprehensive reviews of the literature can be found in [33,61].…”