Discontinuous Galerkin immersed finite element methods for parabolic interface problems

Yang, Qing; Zhang, Xu

doi:10.1016/j.cam.2015.11.020

Cited by 22 publications

(7 citation statements)

References 30 publications

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“…to solve for u − AL,j−1 and u + AR,j+1 . After Equation ( 20) is solved, both positive and negative values at the two auxiliary points are obtained and ready to be used for approximating either (u + τ ) Γ,j or (u − τ ) Γ,j in Equation (19). The proposed procedure can be used to estimate u + τ and u − τ for complex interfaces.…”

Section: Approximating U +mentioning

confidence: 99%

“…Consequently, carefully designed numerical procedures are required to account for jump conditions (4) in the numerical formulations so that numerical accuracy near the interface can be recovered. Such Cartesian grid interface algorithms have been successfully developed in many finite difference methods [4,6,8,[10][11][12][13][14][15][16][17] and finite element methods [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

“…Comparing with general interface algorithms for parabolic PDEs [6,8,11,[16][17][18][19][20], the interface treatment in the ADI framework is more subtle. This is because the jump conditions (4) are prescribed in the normal direction so that the usual interface treatments naturally couple spatial derivatives in all Cartesian directions.…”

Section: Introductionmentioning

confidence: 99%

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Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

Long

et al. 2021

Computation

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The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.

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Section: Approximating U +mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

Long

et al. 2021

Computation

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“…e unconditionally stable domain decomposition method was obtained by the alternating technique in [25][26][27][28][29][30]. Recently, the numerical methods for parabolic equations have attracted great attention of scholars [31][32][33][34][35][36][37][38][39][40][41][42]. Meanwhile, the finite volume element methods for elliptic problems are studied by Bi et al [43,44], and the finite volume element method for second-order hyperbolic equations is proposed by Chen et al [45].…”

Section: Introductionmentioning

confidence: 99%

The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

Xue

Gong

Feng

2020

Journal of Function Spaces

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In this paper, a splitting Crank–Nicolson (SC-N) scheme with intrinsic parallelism is proposed for parabolic equations. The new algorithm splits the Crank–Nicolson scheme into two domain decomposition methods, each one is applied to compute the values at (n + 1)th time level by use of known numerical solutions at n-th time level, respectively. Then, the average of the above two values is chosen to be the numerical solutions at (n + 1)th time level. The new algorithm obtains accuracy of the Crank–Nicolson scheme while maintaining parallelism and unconditional stability. This algorithm can be extended to solve two-dimensional parabolic equations by alternating direction implicit (ADI) technique. Numerical experiments illustrate the accuracy and efficiency of the new algorithm.

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“…Various specially designed numerical schemes with the jump conditions being incorporated into discretization have been introduced in the literature for solving parabolic interface problems. These include finite element methods and finite volume methods based on body-fitted interface treatments [3,8,32,33,34], finite difference methods based on Cartesian grids [2,5,6,18,19,22,17,30,40,24,36,20], and immersed finite element methods based on Cartesian meshes [25,39]. For instance, the immersed interface method (IIM) is one of the most successful finite difference methods in solving parabolic interface problems [5,6,18,19,20].…”

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confidence: 99%

A multigrid based finite difference method for solving parabolic interface problem

Feng

Zhao

2021

era

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In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.

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Discontinuous Galerkin immersed finite element methods for parabolic interface problems

Cited by 22 publications

References 30 publications

Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

The Splitting Crank–Nicolson Scheme with Intrinsic Parallelism for Solving Parabolic Equations

A multigrid based finite difference method for solving parabolic interface problem

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