231 research outputs found
A nonconforming immersed finite element method for elliptic interface problems
A new immersed finite element (IFE) method is developed for second-order
elliptic problems with discontinuous diffusion coefficient. The IFE space is
constructed based on the rotated Q1 nonconforming finite elements with the
integral-value degrees of freedom. The standard nonconforming Galerkin method
is employed in this IFE method without any penalty stabilization term. Error
estimates in energy and L2 norms are proved to be better than and , respectively, where the logarithm factors reflect
jump discontinuity. Numerical results are reported to confirm our analysis
An Analysis of broken -Nonconforming Finite Element Method For Interface Problems
We study some numerical methods for solving second order elliptic problem
with interface. We introduce an immersed interface finite element method based
on the `broken' -nonconforming piecewise linear polynomials on interface
triangular elements having edge averages as degrees of freedom. This linear
polynomials are broken to match the homogeneous jump condition along the
interface which is allowed to cut through the element. We prove optimal orders
of convergence in and -norm. Next we propose a mixed finite volume
method in the context introduced in \cite{Kwak2003} using the Raviart-Thomas
mixed finite element and this `broken' -nonconforming element. The
advantage of this mixed finite volume method is that once we solve the
symmetric positive definite pressure equation(without Lagrangian multiplier),
the velocity can be computed locally by a simple formula. This procedure avoids
solving the saddle point problem. Furthermore, we show optimal error estimates
of velocity and pressure in our mixed finite volume method. Numerical results
show optimal orders of error in -norm and broken -norm for the
pressure, and in H(\Div)-norm for the velocity
A stabilized immersed finite element method for the interface elasticity problems
We develop a new finite element method for solving planar elasticity problems
involving of heterogeneous materials with a mesh not necessarily aligning with
the interface of the materials. This method is based on the `broken'
Crouzeix-Raviart -nonconforming finite element method for elliptic
interface problems \cite{Kwak-We-Ch}.
To ensure the coercivity of the bilinear form arising from using the
nonconforming finite elements, we add stabilizing terms as in the discontinuous
Galerkin (DG) method \cite{Arnold-IP},\cite{Ar-B-Co-Ma},\cite{Wheeler}. The
novelty of our method is that we use meshes independent of the interface, so
that the interface may cut through the elements. Instead, we modify the basis
functions so that they satisfy the Laplace-Young condition along the interface
of each element. We prove optimal and divergence norm error estimates.
Numerical experiments are carried out to demonstrate that the our method is
optimal for various Lam\`e parameters and and locking free as
.Comment: Submitted to M2an on May 18 2015. Added a new author (Dae H. Kyeong
Immersed Finite Element Method for Eigenvalue Problem
We consider the approximation of elliptic eigenvalue problem with an immersed
interface. The main aim of this paper is to prove the stability and convergence
of an immersed finite element method (IFEM) for eigenvalues using
Crouzeix-Raviart -nonconforming approximation. We show that spectral
analysis for the classical eigenvalue problem can be easily applied to our
model problem. We analyze the IFEM for elliptic eigenvalue problem with an
immersed interface and derive the optimal convergence of eigenvalues. Numerical
experiments demonstrate our theoretical results
A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems
In this paper, a stabilized extended finite element method is proposed for
Stokes interface problems on unfitted triangulation elements which do not
require the interface align with the triangulation. The velocity solution and
pressure solution on each side of the interface are separately expanded in the
standard nonconforming piecewise linear polynomials and the piecewise constant
polynomials, respectively. Harmonic weighted fluxes and arithmetic fluxes are
used across the interface and cut edges (segment of the edges cut by the
interface), respectively. Extra stabilization terms involving velocity and
pressure are added to ensure the stable inf-sup condition. We show a priori
error estimates under additional regularity hypothesis. Moreover, the errors
{in energy and norms for velocity and the error in norm for
pressure} are robust with respect to the viscosity {and independent of the
location of the interface}. Results of numerical experiments are presented to
{support} the theoretical analysis.Comment: 36 page
An Immersed Weak Galerkin Method For Elliptic Interface Problems
In this paper, we present an immersed weak Galerkin method for solving
second-order elliptic interface problems. The proposed method does not require
the meshes to be aligned with the interface. Consequently, uniform Cartesian
meshes can be used for nontrivial interfacial geometry. We show the existence
and uniqueness of the numerical algorithm, and prove the error estimates for
the energy norm. Numerical results are reported to demonstrate the performance
of the method
An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid
We develop a numerical scheme for a two-phase immiscible flow in
heterogeneous porous media using a structured grid finite element method, which
have been successfully used for the computation of various physical
applications involving elliptic equations \cite{li2003new, li2004immersed,
chang2011discontinuous, chou2010optimal, kwak2010analysis}. The proposed method
is based on the implicit pressure-explicit saturation procedure. To solve the
pressure equation, we use an IFEM based on the Rannacher-Turek
\cite{rannacher1992simple} nonconforming space, which is a modification of the
work in \cite{kwak2010analysis} where `broken' nonconforming element of
Crouzeix-Raviart \cite{crouzeix1973conforming} was developed.
For the Darcy velocity, we apply the mixed finite volume method studied in
\cite{chou2003mixed, kwak2010analysis} on the basis of immersed finite element
method (IFEM). In this way, the Darcy velocity of the flow can be computed
cheaply (locally) after we solve the pressure equation. The computed Darcy
velocity is used to solve the saturation equation explicitly. Thus the whole
procedure can be implemented in an efficient way using a structured grid which
is independent of the underlying heterogeneous porous media. Numerical results
show that our method exhibits optimal order convergence rates for the pressure
and velocity variables, and suboptimal rate for saturation
A Priori Error Estimates for Some Discontinuous Galerkin Immersed Finite Element Methods
In this paper, we derive a priori error estimates for a class of interior
penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE)
functions for a classic second-order elliptic interface problem. The error
estimation shows that these methods can converge optimally in a mesh-dependent
energy norm. The combination of IFEs and DG formulation in these methods allows
local mesh refinement in the Cartesian mesh structure for interface problems.
Numerical results are provided to demonstrate the convergence and local mesh
refinement features of these DG-IFE methods
Approximation Capabilities of Immersed Finite Element Spaces for Elasticity Interface Problems
We construct and analyze a group of immersed finite element (IFE) spaces
formed by linear, bilinear and rotated Q1 polynomials for solving planar
elasticity equation involving interface. The shape functions in these IFE
spaces are constructed through a group of approximate jump conditions such that
the unisolvence of the bilinear and rotated Q1 IFE shape functions are always
guaranteed regardless of the Lam\`e parameters and the interface location. The
boundedness property and a group of identities of the proposed IFE shape
functions are established. A multi-point Taylor expansion is utilized to show
the optimal approximation capabilities for the proposed IFE spaces through the
Lagrange type interpolation operators
Immersed finite element method for eigenvalue problems in elasticity
We consider the approximation of eigenvalue problems for elasticity equations
with interface. This kind of problems can be efficiently discretized by using
immersed finite element method (IFEM) based on Crouzeix-Raviart
P1-nonconforming element. The stability and the optimal convergence of IFEM for
solving eigenvalue problems with interface are proved by adapting spectral
analysis methods for the classical eigenvalue problem. Numerical experiments
demonstrate our theoretical results.Comment: 17 pages, 11 figures, 1 tabl
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