Solvers for the eikonal equation fall into one of two camps: 1) direct solvers, which are typically based on Dijkstra's algorithm, the classical example being the fast marching method, and 2) iterative solvers, best represented by the fast sweeping method. Direct solvers run in optimal, output-sensitive time, but are generally only first-order accurate. Iterative methods can be made higher-order, but require wide stencils; additionally, their performance degrades when applied to complicated domains. We develop high-order, semi-Lagrangian direct solvers for the eikonal equation, which use compact stencils. These solvers march the partial derivatives of the eikonal up to some order in addition to the eikonal itself. Semi-Lagrangian updates approximate the eikonal and updating characteristics using local Hermite interpolants. We present convergence guarantees, thorough numerical results, and numerical examples related to room acoustic modeling, which is our motivating application.
Bilayer plates are slender structures made of two thin layers of different materials glued together. These layers react differently to non-mechanical stimuli and develop large bending deformations. We design a LDG approach for the constrained minimization problem of bilayer plates. With this new discretization, we prove the $\Gamma$-convergence and design a fully practical gradient flow scheme. We also prove the energy stability and the control of constraint defect for this scheme.
We present a novel second-order scheme for gradient flows of the Dirichlet energy of maps from a three-dimensional domain to the unit sphere. The unit-length constraint is treated with help of a Lagrange multiplier, and Dirichlet conditions are imposed weakly on piecewise quadratic level sets defined on an unfitted regular mesh. The method is applied to study trapping pits for colloid particles immersed in a liquid crystal.