Related papers: A diffuse interface approach for vector-valued PDE…
We study multilevel techniques, commonly used in PDE multigrid literature, to solve structured optimization problems. For a given hierarchy of levels, we formulate a coarse model that approximates the problem at each level and provides a…
We devise a numerical method for passive advection of a surface, such as the interface between two incompressible fluids, across a computational mesh. The method is called isoAdvector, and is developed for general meshes consisting of…
We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from the commutation of 2-dimensional vector fields. We also solve the associated Cauchy problems, using the recently developed Inverse Scattering Transform for…
We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the…
An approach to the equivalence problem of vector valued maps is offered which, in particular, covers the equivalence problem of paths and patches of differential geometry with respect to different motion groups. In the last case, in…
In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a…
This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes. We propose to discretize the interface via triangle edge cuts which represent the intersections between the interface and the triangle…
We present a velocity-based moving mesh virtual element method for the numerical solution of PDEs involving moving boundaries. The virtual element method is used for computing both the mesh velocity and a conservative Arbitrary…
A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background…
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary…
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…
Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force…
An investigation of the effect of surface diffusion in random deposition model is made by analytical methods and reasoning. For any given site, the extent to which a particle can diffuse is decided by the morphology in the immediate…
The hybrid high-order method is a modern numerical framework for the approximation of elliptic PDEs. We present here an extension of the hybrid high-order method to meshes possessing curved edges/faces. Such an extension allows us to…
This paper presents a mathematical analysis of an elliptic partial differential equation (PDE) designed to compute the geometric thickness of a given shape. The PDE-based formulation provides a direct and systematic approach to evaluate…
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by…
We present a new method for performing Boolean operations on volumes represented as triangle meshes. In contrast to existing methods which treat meshes as 3D polyhedra and try to partition the faces at their exact intersection curves, we…
In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded…