Related papers: A Practical Phase Field Method for an Elliptic Sur…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
Partial differential equations posed on surfaces arise in a number of applications. In this survey we describe three popular finite element methods for approximating solutions to the Laplace-Beltrami problem posed on an $n$-dimensional…
Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are…
Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et.al. (SIAM Journal on Scientific Computing 45(4)(2023)) and as an improvement, a…
This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in…
The phase field approach to modeling fracture uses a diffuse damage field to represent a crack. This addresses the singularities that arise at the crack tip in computations with sharp interface models, mollifying some of the difficulties…
We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of…
The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order…
Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately…
Partial differential equations (PDEs) on surfaces appear in many applications throughout the natural and applied sciences. The classical closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook…
A multi-scale meshfree particle method for macroscopic mean field approximations of generalized interacting particle models is developed and investigated. The method is working in a uniform way for large and small interaction radii. The…
In this paper, we propose an alternative approach to implement the contact angle boundary condition on immersed surfaces for phase-field simulations of two-phase flows using the Cahn-Hilliard equation on a Cartesian mesh. This simple and…
The level-set method is a popular interface tracking method in two-phase flow simulations. An often-cited reason for using it is that the method naturally handles topological changes in the interface, e.g. merging drops, due to the implicit…
We consider a moving interface that is coupled to an elliptic equation in a heterogeneous medium. The problem is motivated by the study of displacive solid-solid phase transformations. We show that a nearly flat interface is given by the…
We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms…
We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known…
In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique…
We construct a diffuse-interface model of two-phase solidification that quantitatively reproduces the classic free boundary problem on solid-liquid interfaces in the thin-interface limit. Convergence tests and comparisons with boundary…