Related papers: A diffuse interface approach for vector-valued PDE…
We consider a diffuse interface approach for solving an elliptic PDE on a given closed hypersurface. The method is based on a (bulk) finite element scheme employing numerical quadrature for the phase field function and hence is very easy to…
In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable…
Frequently, the design of physicochemical processes requires screening of large numbers of alternative designs with complex geometries. These geometries may result in conformal meshes which introduce stability issues, significant…
Interactions between an evolving solid and inviscid flow can result in substantial computational complexity, particularly in circumstances involving varied boundary conditions between the solid and fluid phases. Examples of such…
In this note we shall introduce a simple, effective numerical method for solving partial differential equations for scalar and vector-valued data defined on surfaces. Even though we shall follow the traditional way to approximate the…
In this paper, we propose an approach for solving PDEs on evolving surfaces using a combination of the trace finite element method and a fast marching method. The numerical approach is based on the Eulerian description of the surface…
Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This…
We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we…
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the…
We study both diffuse and sharp liquid-vapor interfaces. The equilibrium equation of fluids is derived by using the principle of virtual work in a domain including the interfaces. For diffuse interfaces, the surface tension coefficient…
In this paper we propose an efficient second order well balanced finite volume method for modeling complex free surface flows at the aid of a simple diffuse interface method. The employed physical model is a two-phase model derived from the…
This paper is concerned with diffuse-interface approximations of the Willmore flow. We first present numerical results of standard diffuse-interface models for colliding one dimensional interfaces. In such a scenario evolutions towards…
The paper studies a finite element method for computing transport and diffusion along evolving surfaces. The method does not require a parametrization of a surface or an extension of a PDE from a surface into a bulk outer domain. The…
We analyze and test using Fourier extensions that minimize a Hilbert space norm for the purpose of solving partial differential equations (PDEs) on surfaces. In particular, we prove that the approach is arbitrarily high-order and also show…
In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree…
In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the…
We demonstrate a method for filtering images defined on curved surfaces embedded in 3D. Applications are noise removal and the creation of artistic effects. Our approach relies on in-surface diffusion: we formulate Weickert's edge/coherence…
Vectors fields defined on surfaces constitute relevant and useful representations but are rarely used. One reason might be that comparing vector fields across two surfaces of the same genus is not trivial: it requires to transport the…
Multiphase flows are commonly found in chemical engineering processes such as distillation columns, bubble columns, fluidized beds and heat exchangers. The physical boundaries of domains in numerical simulations of multiphase flows are…
The diffuse-domain, or smoothed boundary, method is an attractive approach for solving partial differential equations in complex geometries because of its simplicity and flexibility. In this method the complex geometry is embedded into a…