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The Diffusion of Passive Tracers in Laminar Shear Flow

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SOLVING PDES IN COMPLEX GEOMETRIES: A DIFFUSE DOMAIN APPROACH.

X Li¹, J Lowengrub, A Rätz

¹Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875, USA ( xli@math.uci.edu ).

Communications in Mathematical Sciences

|September 28, 2011

Summary

This summary is machine-generated.

This study introduces a novel method for solving partial differential equations in complex geometries. The approach uses a diffuse domain to handle various boundary conditions, enabling accurate simulations for moving and growing structures.

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Area of Science:

Computational Mathematics
Numerical Analysis
Applied Physics

Background:

Solving partial differential equations (PDEs) in complex geometries is challenging.
Existing methods often struggle with stationary or moving boundaries and diverse boundary conditions.

Purpose of the Study:

To present a general and robust approach for solving PDEs in complex, stationary, or moving geometries.
To accommodate Dirichlet, Neumann, and Robin boundary conditions effectively.

Main Methods:

Implicitly represent geometry using a phase field function, creating a diffuse domain.
Reformulate the PDE on a larger, regular domain with additional lower-order terms for boundary conditions.
Employ matched asymptotic expansions to prove convergence of solutions.

Main Results:

The reformulated PDE maintains the original order with added terms approximating boundary conditions.
Numerical simulations confirm the convergence of solutions to the original problem.
The method is successfully applied to growing domains and complex 3D structures.

Conclusions:

The diffuse domain method offers a versatile solution for PDEs in complex geometries.
This approach has significant potential applications in fields like cell biology and heteroepitaxy.