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The immersed interface method for the Navier–Stokes equations with singular forces
 J. Comput. Phys
"... Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, t ..."
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Cited by 81 (5 self)
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Peskin’s Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be firstorder accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a secondorder projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully secondorder accuracy for the velocity and nearly secondorder accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. c ○ 2001 Academic Press Key Words: Navier–Stokes equations; interface; discontinuous and nonsmooth solution; immersed interface method; immersed boundary method; projection method; level set method. 1.
A Hybrid Method for Moving Interface Problems with Application to the HeleShaw Flow
, 1997
"... this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to t ..."
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Cited by 81 (22 self)
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this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a secondorder Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This newapproach is then applied to HeleShaw flow, an unstable flow involving two fluids with very different viscosity. 1997 Academic Press L
Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems
 Numer. Math
, 1996
"... In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of ar ..."
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Cited by 78 (11 self)
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In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Mathematics Subject Classification (1991): 65N30, 65F10. A running title: Finite element methods for interface problems. Correspondence to: Dr. Jun Zou Email: zou@math.cuhk.edu.hk Fax: (852) 2603 5154 1 Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. Email: zmchen@math03.math.ac.cn. The work of this author was partially supported by China National Natural Science Foundation. 2 Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zou@math.cuhk....
A Fast Iterative Algorithm For Elliptic Interface Problems
 SIAM J. Numer. Anal
, 1995
"... . A fast, second order accurate iterative method is proposed for the elliptic equation r \Delta (fi(x; y)ru) = f(x; y) in a rectangular region\Omega in 2 space dimensions. We assume that there is an irregular interface across which the coefficient fi, the solution u and its derivatives, and/or the ..."
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Cited by 71 (20 self)
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. A fast, second order accurate iterative method is proposed for the elliptic equation r \Delta (fi(x; y)ru) = f(x; y) in a rectangular region\Omega in 2 space dimensions. We assume that there is an irregular interface across which the coefficient fi, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the jump in fi is large. The interface may or may not align with a underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [SINUM, 4 (1994), pp. 10191044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second order difference scheme for a corresponding Poisson equation in the region, and a second order discretization for a Ne...
Numerical Approximations of Singular Source Terms in Differential Equations
 J. Comput. Phys
, 2003
"... Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the ..."
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Cited by 52 (3 self)
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Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi dimensions. New high order multi dimensional techniques for di#erential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions to extend one dimensional regularization to higher dimensions may produce O(1) errors.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
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A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 41 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious
Analysis of Stiffness in the Immersed Boundary Method and Implications for Timestepping Schemes
 J. Comput. Phys
, 1998
"... The immersed boundary method is known to exhibit a high degree of numerical stiffness, which is associated with the interaction of immersed elastic fibres with the surrounding fluid. We perform a linear analysis of the underlying equations of motion for immersed fibres, and identify a discrete set ..."
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Cited by 38 (1 self)
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The immersed boundary method is known to exhibit a high degree of numerical stiffness, which is associated with the interaction of immersed elastic fibres with the surrounding fluid. We perform a linear analysis of the underlying equations of motion for immersed fibres, and identify a discrete set of fibre modes which are associated solely with the presence of the fibre. These results are a generalisation of those in a previous paper (SIAM J. Appl. Math., 55(6):15771591, 1995) by including the effect of spreading the singular fibre force over a finite "smoothing radius," which corresponds to the approximate delta function used in the immersed boundary method. We investigate the stability of the fibre modes, their stiffness and dependence on the problem parameters, and the effect that smoothing has on the solution. The analytical results are then extended to include the effects of time discretisation, and conclusions are drawn about the time step restrictions on various expli...