We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.

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Solving pdes in complex geometries: a diffuse domain approach.

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only can these IFE methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the $H^1$-norm and the $L^2$-norm do not deteriorate when the mesh becomes finer, which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods but are also of a great potential to be useful in error analysis for other...

Partially penalized immersed finite element methods for elliptic interface problems

Abstract. This paper is to develop immersed finite element (IFE) functions for solving second order elliptic boundary value problems with discontinuous coefficients and non-homogeneous jump conditions. These IFE functions can be formed on meshes independent of interface. Numerical examples demonstrate that these IFE functions have the usual approximation capability expected from polynomials employed. The related IFE methods based on the Galerkin formulation can be considered as natural extensions of those IFE methods in the literature developed for homogeneous jump conditions, and they can optimally solve the interface problems with a nonhomogeneous flux jump condition.

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Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions

An interface-fitted mesh generator and virtual element methods for elliptic interface problems ☆

In this article an unfitted discontinuous Galerkin method is proposed to discretize elliptic interface problems. The method is based on the symmetric interior penalty discontinuous Galerkin method and can also be interpreted as a generalization of the method given by A. Hansbo and P. Hansbo [Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537--5552]. We prove the optimal $h$-convergence of the method for arbitrary $p$ in the energy- and in the $L_2$-norm. In fact we present an $hp$-error estimate. We introduce a variant of the method, for which the same $hp$-convergence rates can be proved in the energy norm and which aims at controlling pointwise errors near the interface. The analysis includes grids with hanging nodes, and the proposed method is symmetric and inherits the attractive locality property of general discontinuous Galerkin methods. Numerical experiments are included.

An Unfitted Discontinuous Galerkin Method Applied to Elliptic Interface Problems

In this work, a class of new finite-element methods, called immersed-interface finite-element methods, is developed to solve elliptic interface problems with nonhomogeneous jump conditions. Simple non-body-fitted meshes are used. A single function that satisfies the same nonhomogeneous jump conditions is constructed using a level-set representation of the interface. With such a function, the discontinuities across the interface in the solution and flux are removed, and an equivalent elliptic interface problem with homogeneous jump conditions is formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require remeshing.

Yan Gong

Papers

Immersed-Interface Finite-Element Methods for Elliptic Interface Problems with Nonhomogeneous Jump Conditions

Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions

An isogeometric boundary element method for transient heat transfer problems in inhomogeneous materials and the non-iterative inversion of loads