Method For Interface Problems Research Articles

A weighted Nitsche’s method for interface problems with higher-order simplex elements

A weighted Nitsche’s method for interface problems with higher-order simplex elements

A partially penalty immersed Crouzeix-Raviart finite element method for interface problems

The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the L^{2} norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.

A binary level set model and some applications to Mumford-Shah image segmentation

In this paper, we propose a PDE-based level set method. Traditionally, interfaces are represented by the zero level set of continuous level set functions. Instead, we let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e., it can only be 1 or -1; thus, our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.

Nitsche's method for interface problems in computa‐tional mechanics

AbstractWe give a review of Nitsche's method applied to interface problems, involving real or artificial interfaces. Applications to unfitted meshes, Chimera meshes, cut meshes, fictitious domain methods, and model coupling are discussed. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations

Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov Galerkin finite-element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary value problem for the sensitivities. In particular, the sensitivity boundary value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.

Discussion of “ Finite Element Method for Interface Problems ” by Santosh K. Arya and Gilbert A. Hegemier (February, 1982)

Discussion of “ <i>Finite Element Method for Interface Problems</i> ” by Santosh K. Arya and Gilbert A. Hegemier (February, 1982)