Finite elements on evolving surfaces

By Gerhard Dziuk, Charles M. Elliott. 2007

400 citations recorded by the source corpus.

In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time. The key idea is based on approximating (t) by an evolving interpolated polyhedral (polygonal if n = 1) surface h(t) consisting of a union of simplices (triangles for n = 2) whose vertices lie on (t). A finite-element space of functions is then defined by taking the set of all continuous functions on h(t) which are linear affine on each simplex. The finite-element nodal basis functions enjoy a transport property which simplifies the computation. We formulate a conservation law for a scalar quantity on (t) and, in the case of a diffusive flux, derive a transport and diffusion equation which takes into account the tangential velocity and the local stretching of the surface. Using surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations on (t). Our finite-element method is applied to the weak form of the conservation equation. The computations of the mass and element stiffness matrices are simple and straightforward. Error bounds are derived in the case of semi-discretization in space. Numerical experiments are described which indicate the order of convergence and also the power of the method. We describe how this framework may be employed in applications. Key Words: finite elements; evolving surfaces; conservation; diffusion; existence; error estimates; computations

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a68c7ef246ac921c3c8630dcf0330de684efe3ef

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Finite element analysis for a coupled bulk-surface partial differential equation (2013, methodology): “…nloaded from https://academ ic.oup.com /im ajna/article-abstract/33/2/377/653348 by guest on 28 D ecem ber 2018 1.2 Applications In recent times there has been a great deal of attention paid to problems involving diffusion on a surface, for example, Dziuk & Elliott (2007b) and references therein.”
Modelling fluid deformable surfaces with an emphasis on biological interfaces (2018, background): “Even if the interaction with a bulk fluid is not considered, the interfacial velocity can be extended to a tubular neighbourhood in 3D around the interface to find the governing equations, which can be shown to be independent of the extension (Dziuk & Elliott 2007, 2013).”
Backward difference time discretization of parabolic differential equations on evolving surfaces (2013, neutral): “For the reader’s convenience we recall some technical preliminaries from [3].”
Finite element methods for surface PDEs* (2013, methodology): “We close this section with an example from Dziuk and Elliott (2007a).”