Unfitted Finite Element Methods Using Bulk Meshes for Surface Partial Differential Equations

Abstract

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ R n+1 , is embedded in a polyhedral domain in R n+1 consisting of a union, T h , of (n + 1)-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on T h . Our first method is a sharp interface method, SIF, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γ h , of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [42]. The second method, NBM, is a narrow band method in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [13] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L 2 and H 1 norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.

Volumetric variational principles for a class of partial differential equations defined on surfaces and curves

“…One way to solve this problem is to introduce a Lagrange multiplier and solve to steady state the following equation: 6) where the multiplier λ is chosen to be…”

Minimal surface computation using a finite element method on an embedded surface

“…(6) Extract zero isosurface points giving n by going over all elements in K n h , interpolating the signed distance function linearly using the tetrahedral basis functions. (7) Compute the velocity field u h n by solving (11). This is carried out by solving the matrix equation…”

“…Instead of using a global phase field, MINIMAL SURFACE COMPUTATION ON AN EMBEDDED SURFACE 503 diffuse interface methods have been devised for the solution of surface differential equations, e.g., as localized phase fields as in Rätz and Voigt [7], as closest point embeddings, described by Ruuth and Merriman [8], or as in the narrow band finite element approximation using level set techniques discussed in Deckelnick et al [9]; see also Dziuk and Elliot [10]. Finally, we mention the recently developed sharp interface method by Deckelnick, Elliott, and Ranner [11], which is closely related to the work in [2].…”

A granular computing method for nonlinear convection-diffusion equation

“…Therefore, numerical methods must be applied. At present, the major numerical methods are finite element methods(FEM) [2][3][4] [5], finite difference methods (FDM) [6] [7], finite volume methods(FVM) [8][9] [10]. However, when the FEM or FDM are used to solve the NCDE, it exhibits excessive numerical diffusion and nonphysical oscillation [11].…”