A finite element method for surface PDEs: matrix properties

A finite element method for surface PDEs: matrix properties

We consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces. Themain idea of this method is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a couplin...

A Multiscale Finite Element Method for Pdes with Oscillatory Coefficients1

We present a multiscale finite element method (MFEM) for solving partial differential equations with highly oscillatory coefficients. MFEM provides a general and systematic framework for effectively capturing the large scale solutions without directly resolving the small scale details. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the loc...

A Volume Mesh Finite Element Method for Pdes on Surfaces

We treat a surface finite element method that is based on the trace of a standard finite element space on a tetrahedral triangulation of an outer domain that contains a stationary 2D surface. This surface FEM is used to discretize partial differential equation on the surface. We demonstrate the performance of this method for stationary and time-dependent diffusion equations. For the stationary ...

The Finite Element Method for 2D elliptic PDEs

Eulerian finite element method for parabolic PDEs on implicit surfaces

We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface Γ contained in a domain Ω ⊂ Rn+1. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function Φ whose zero level set is Γ . Eulerian surface gradients are formulated by using a projection of the gradient in Rn+1 onto the level ...