A relaxed weak Galerkin method for elliptic interface problems with low regularity

Song, Lunji; Zhao, Shan; Liu, Kaifang

doi:10.1016/j.apnum.2018.01.021

Cited by 12 publications

(8 citation statements)

References 28 publications

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“…In this section, we shall describe the weak Galerkin finite element discretization for the problem (1.1)–(1.3) and review the definition of the weak gradient operator. Previous works on weak Galerkin methods for elliptic interface problems can be found in .…”

Section: Preliminaries and Weak Galerkin Discretizationmentioning

confidence: 99%

“…Then the optimal order error estimates in both H 1 and L 2 norms are established with lowest order WG finite element space

({scriptP}_{k} (K), {scriptP}_{k - 1} (\partial K), {([], P_{k - 1} ((), K))}^{2})

in . More recently, a relaxed weak Galerkin method for elliptic interface problems has been proposed in . Like LDG methods, weak Galerkin algorithm presented in does not require special treatment for the discontinuity across the interface.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, a relaxed weak Galerkin method for elliptic interface problems has been proposed in . Like LDG methods, weak Galerkin algorithm presented in does not require special treatment for the discontinuity across the interface. Both interface jump functions are enforced in a natural framework.…”

Section: Introductionmentioning

confidence: 99%

“…Convergence analysis has been carried out in L p ‐norm for WG finite element space

({scriptP}_{k} (K), {scriptP}_{k} (\partial K), {([], P_{k - 1} ((), K))}^{2})

. Extension of the WG algorithm presented in to time‐dependent interface problems would be very interesting. In this article, we extend the work of to the electric interface model (1.1)–(1.3).…”

Section: Introductionmentioning

confidence: 99%

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Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

Deka

Roy

2019

Numerical Methods Partial

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In this paper, the weak Galerkin finite element method (WG-FEM) is applied to a pulsed electric model arising in biological tissue when a biological cell is exposed to an electric field. A fitted WG-FEM is proposed to approximate the voltage of the pulsed electric model across the physical media involving an electric interface (surface membrane), and heterogeneous permittivity and a heterogeneous conductivity. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Optimal pointwise-in-time error estimates in L 2 -norm and H 1 -norm are shown to hold for the semidiscrete scheme even if the regularity of the solution is low on the whole domain. Furthermore, a fully discrete approximation based on backward Euler scheme is analyzed and related optimal error estimates are derived. KEYWORDSinterface, optimal error estimates, pulsed electric field, weak Galerkin method 1

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Section: Preliminaries and Weak Galerkin Discretizationmentioning

confidence: 99%

“…Then the optimal order error estimates in both H 1 and L 2 norms are established with lowest order WG finite element space

({scriptP}_{k} (K), {scriptP}_{k - 1} (\partial K), {([], P_{k - 1} ((), K))}^{2})

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Convergence analysis has been carried out in L p ‐norm for WG finite element space

({scriptP}_{k} (K), {scriptP}_{k} (\partial K), {([], P_{k - 1} ((), K))}^{2})

Section: Introductionmentioning

confidence: 99%

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Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

Deka

Roy

2019

Numerical Methods Partial

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“…Consider the polygon interface problem with low regularity to illustrating the capability of OPWG. The example was considered in[19].Let the domain Ω = (0, 1) 2 with Ω 1 = [0.2, 0.8] × [0.3, 0.7] and Ω 2 = Ω/Ω 1 . The piecewise diffusive coefficients are taken as A 11 = 80, and A 22 = 1, and the exact solutions are chosen as follows: u(x, y) = (x 2 + y 2 ) 1.5 + sin(x + y), if (x, y) ∈ Ω 1 , v(x, y) = 1 + [(x − 0.5) 2 + (y − 0.5) 2 ] 0.5 , if (x, y) ∈ Ω 2 .…”

mentioning

confidence: 99%

A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems

Song¹,

Qi²,

Liu³

et al. 2021

DCDS-B

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We introduce an over-penalized weak Galerkin method for elliptic interface problems with non-homogeneous boundary conditions and discontinuous coefficients, where the method combines a weak Galerkin stabilizer with interior penalty terms. This method employs double-valued functions on interior edges of elements instead of single-valued ones and elements (P k , P k , [P k−1 ] 2). As an advantage of the method, elliptic interface problems with low regularity are approximated well. The over-penalized weak Galerkin method is based on weak functions whose edge part is double-valued on each interior edge sharing by two neighboring elements. Jumps between the edge parts are naturally used to define penalty terms. The over-penalized weak Galerkin method allows to use arbitrary meshes, even for low regularity solutions. These features make the new method more flexible and efficient for solving interface equations. Furthermore, a priori error estimates in energy and L 2 norms are derived rigorously, and numerical results confirm the effectiveness of the method.

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A systematic study on weak Galerkin finite element method for second‐order parabolic problems

Deka

Кумар

2022

Numerical Methods Partial

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In the present work, we have described a systematic numerical study on weak Galerkin (WG) finite element method for second‐order linear parabolic problems by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in L∞()L2$$ {L}^{\infty}\left({L}^2\right) $$ and L∞()H1$$ {L}^{\infty}\left({H}^1\right) $$ norms for a general WG element 𝒫kfalse(Kfalse),𝒫jfalse(∂Kfalse),[]𝒫l(K)2, where k≥1$$ k\ge 1 $$, j≥0$$ j\ge 0 $$ and l≥0$$ l\ge 0 $$ are arbitrary integers. The fully discrete space–time discretization is based on a first order in time Euler scheme. Numerical experiments are reported to justify the robustness, reliability and accuracy of the WG finite element method.

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A relaxed weak Galerkin method for elliptic interface problems with low regularity

Cited by 12 publications

References 28 publications

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

Weak Galerkin finite element methods for electric interface model with nonhomogeneous jump conditions

A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems

A systematic study on weak Galerkin finite element method for second‐order parabolic problems

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