Simplified immersed interface methods for elliptic interface problems with straight interfaces

Feng, Xiufang; Li, Zhilin

doi:10.1002/num.20614

Cited by 12 publications

(6 citation statements)

References 27 publications

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“…The exact solution is u ( r , θ ) = r cos( θ ), so that

f (r, θ) = \{\begin{matrix} center \\ w_{-}^{2} r \cos (θ), (r, θ) \in [1,1.5] \times (0, 2 π), \\ w_{+}^{2} r \cos (θ), (r, θ) \in (1.5,2] \times (0, 2 π) . \end{matrix}

For the matrices Λ and

Z

for periodic boundary conditions, see [7]. The errors and convergence rates are given in Tables 11–13. Example [2, Example 4.2] We consider

β (u_{italicxx} (x, y) + u_{italicyy} (x, y)) + σ (x) u (x, y) = f (x, y), (x, y) \in Ω \equiv (- 1, 1) \times (- 1, 1),

with boundary conditions

u (- 1, y) = e^{- 1} [\sin (y) + y^{2}], u (1, y) = - 1 - y

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Example [2, Example 4.4] We consider the following problem;

β (u_{italicxx} + u_{italicyy}) + σ (x) u (x, y) = f (x, y), (x, y) \in Ω \equiv (- 1, 1) \times (- 1, 1),

with boundary conditions

u (- 1, y) = e^{- 1} [\sin (y) + y^{2}], u (1, y) = e^{1} [\sin (y) + y^{2}], y \in [- 1, 1],

and

u (x, - 1) = e^{x} [x^{2} \sin (- 1) + 1], u (x, 1) = e^{x} [x^{2} \sin (1) + 1], x \in [- 1, 1],

where

β = \{\begin{matrix} center \\ β_{1}, (x, y) \in [- 1] \end{matrix}

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Such a problem arises in many physical applications, and various numerical techniques have been developed for its approximate solution particularly for the case in which ω 2 , the wave number, is a constant. Recently, high order (i.e., order three or four) compact finite difference methods have been developed for the solution of (1.1) with Dirichlet boundary conditions and a vertical interface at x = x ℓ , which we call Γ ℓ , such that

ω^{2} = \{\begin{matrix} left \\ ω_{-}^{2}, & (x, y) \in [0, x_{ℓ}] \times [0, 1], \\ ω_{+}^{2}, & (x, y) \in (x_{ℓ}, 1] \times [0, 1] . \end{matrix}

In the derivation of these methods, special attention is devoted to the approximation of the interface conditions since, even though u and u x may be continuous across this interface, that is,

{([], u)}_{| x = x_{ℓ}} = {([], u_{x})}_{| x = x_{ℓ}} = 0,

where

{([], u)}_{| x = x_{ℓ}} = \underset{x \to x_{ℓ}^{+}}{normallim} u (x, y) - \underset{x \to x_{ℓ}^{-}}{normallim} u (x, y), y \in (0, 1),

the second or higher order partial derivatives with respect to x and the source function f ( x , y ) may be discontinuous across the interface; see, for example, [2, 3]. Moreover, frequently only Dirichlet boundary conditions are considered avoiding the added complication of deriving high‐order approximations to Neumann boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

Bhal¹,

Danumjaya²,

Fairweather

2020

Numerical Methods Partial

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Orthogonal spline collocation is implemented for the numerical solution of two-dimensional Helmholtz problems with discontinuous coefficients in the unit square. A matrix decomposition algorithm is used to solve the collocation matrix system at a cost of O(N 2 log N) on an N × N partition of the unit square. The results of numerical experiments demonstrate the efficacy of this approach, exhibiting optimal global estimates in various norms and superconvergence phenomena for a broad spectrum of wave numbers.

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“…The exact solution is u ( r , θ ) = r cos( θ ), so that

f (r, θ) = \{\begin{matrix} center \\ w_{-}^{2} r \cos (θ), (r, θ) \in [1,1.5] \times (0, 2 π), \\ w_{+}^{2} r \cos (θ), (r, θ) \in (1.5,2] \times (0, 2 π) . \end{matrix}

For the matrices Λ and

Z

for periodic boundary conditions, see [7]. The errors and convergence rates are given in Tables 11–13. Example [2, Example 4.2] We consider

β (u_{italicxx} (x, y) + u_{italicyy} (x, y)) + σ (x) u (x, y) = f (x, y), (x, y) \in Ω \equiv (- 1, 1) \times (- 1, 1),

with boundary conditions

u (- 1, y) = e^{- 1} [\sin (y) + y^{2}], u (1, y) = - 1 - y

…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Example [2, Example 4.4] We consider the following problem;

β (u_{italicxx} + u_{italicyy}) + σ (x) u (x, y) = f (x, y), (x, y) \in Ω \equiv (- 1, 1) \times (- 1, 1),

with boundary conditions

u (- 1, y) = e^{- 1} [\sin (y) + y^{2}], u (1, y) = e^{1} [\sin (y) + y^{2}], y \in [- 1, 1],

and

u (x, - 1) = e^{x} [x^{2} \sin (- 1) + 1], u (x, 1) = e^{x} [x^{2} \sin (1) + 1], x \in [- 1, 1],

where

β = \{\begin{matrix} center \\ β_{1}, (x, y) \in [- 1] \end{matrix}

…”

Section: Numerical Resultsmentioning

confidence: 99%

ω^{2} = \{\begin{matrix} left \\ ω_{-}^{2}, & (x, y) \in [0, x_{ℓ}] \times [0, 1], \\ ω_{+}^{2}, & (x, y) \in (x_{ℓ}, 1] \times [0, 1] . \end{matrix}

In the derivation of these methods, special attention is devoted to the approximation of the interface conditions since, even though u and u x may be continuous across this interface, that is,

{([], u)}_{| x = x_{ℓ}} = {([], u_{x})}_{| x = x_{ℓ}} = 0,

where

{([], u)}_{| x = x_{ℓ}} = \underset{x \to x_{ℓ}^{+}}{normallim} u (x, y) - \underset{x \to x_{ℓ}^{-}}{normallim} u (x, y), y \in (0, 1),

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

Bhal¹,

Danumjaya²,

Fairweather

2020

Numerical Methods Partial

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show abstract

“…Interface problems have attracted a lot of attention from both theoretical analysis and numerical studies over the years . Many numerical methods such as the conventional finite element method designed for smooth solutions do not work, or work poorly, for interface problems .…”

Section: Introductionmentioning

confidence: 99%

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

Taleei

Dehghan

2014

Numerical Methods Partial

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We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong-form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second-order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well-known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher-order derivatives using the approximation of lower-order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second-and higher-orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems.

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“…Therefore, different numerical methods on Cartesian mesh have been developed for interface problems, including immersed boundary method [78][79][80], immersed interface method [81][82][83], matched interface and boundary methods [84,85], cut-cell methods [86,87], embedded boundary methods [88], Cartesian grid methods [89,90], generalized finite element methods [91][92][93][94], extended finite element methods [95][96][97], and Eulerian-Lagrangian localized adjoint methods [98][99][100].…”

Section: Introductionmentioning

confidence: 99%

A selective immersed discontinuous Galerkin method for elliptic interface problems

Lin

2013

Math. Meth. Appl. Sci.

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This article proposes a selective immersed discontinuous Galerkin method based on bilinear immersed finite elements (IFE) for solving second‐order elliptic interface problems. This method applies the discontinuous Galerkin formulation wherever selected, such as those elements around an interface or a singular source, but the regular Galerkin formulation everywhere else. A selective bilinear IFE space is constructed and applied to the selective immersed discontinuous Galerkin method based on either the symmetric or nonsymmetric interior penalty discontinuous Galerkin formulation. The new method can solve an interface problem by a rectangular mesh with local mesh refinement independent of the interface even if its geometry is nontrivial. Meanwhile, if desired, its computational cost can be maintained very close to that of the standard Galerkin IFE method. It is shown that the selective bilinear IFE space has the optimal approximation capability expected from piecewise bilinear polynomials. Numerical examples are provided to demonstrate features of this method, including the effectiveness of local mesh refinement around the interface and the sensitivity to the penalty parameters. Copyright © 2013 John Wiley & Sons, Ltd.

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Simplified immersed interface methods for elliptic interface problems with straight interfaces

Cited by 12 publications

References 27 publications

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

A fourth–order orthogonal spline collocation method for two‐dimensional Helmholtz problems with interfaces

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

A selective immersed discontinuous Galerkin method for elliptic interface problems

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