Finite element methods for surface PDEs
Gerhard Dziuk,Charles M. Elliott +1 more
TLDR
Finite element methods for approximating the solution of partial differential equations on surfaces are considered, focusing on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods.Abstract:
In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.read more
Citations
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Journal ArticleDOI
Physically based modeling in catchment hydrology at 50: Survey and outlook
Claudio Paniconi,Mario Putti +1 more
TL;DR: An historical overview of some of the key developments in physically based hydrological modeling is given, emphasizing how the interplay between theory, experiments, and modeling has contributed to advancing the state of the art.
Journal ArticleDOI
A coupled surface-Cahn--Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes
TL;DR: In this paper, a model for lipid raft formation and dynamics in biological membranes is proposed and investigated, where the lipid composition of the membrane and an interaction with cholesterol are modeled in the form of an extended Cahn-Hilliard equation.
Book ChapterDOI
The phase field method for geometric moving interfaces and their numerical approximations
Qiang Du,Xiaobing Feng +1 more
TL;DR: In this article, the authors present a holistic overview about the main ideas of phase field modelling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state of the art of numerical approximations of various phase field models with an emphasis on discussing the main idea of numerical analysis techniques.
Journal ArticleDOI
Incompressible fluid problems on embedded surfaces: Modeling and variational formulations
TL;DR: In this article, the surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space, and the governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics.
Journal ArticleDOI
A stabilized cut finite element method for partial differential equations on surfaces: The Laplace–Beltrami operator
TL;DR: In this article, the Laplace-Beltrami problem on a smooth two-dimensional surface embedded into a three-dimensional space meshed with tetrahedra is considered.
References
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Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Journal ArticleDOI
The Chemical Basis of Morphogenesis
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Book
The Finite Element Method for Elliptic Problems
Philippe G. Ciarlet,J. T. Oden +1 more
TL;DR: The finite element method has been applied to a variety of nonlinear problems, e.g., Elliptic boundary value problems as discussed by the authors, plate problems, and second-order problems.
Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.