A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

Song, Lunji; Liu, Kaifang; Zhao, Shan

doi:10.1007/s10915-016-0296-4

Cited by 6 publications

(5 citation statements)

References 11 publications

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“…In order to show an error estimate in L 2 norm of (9), we apply the Nitsche's trick. Consider the following auxiliary problem of (1)-(4) (see [3] and [18]), i.e. to find w ∈ H 1 0 (Ω) satisfy      −∇ • (A∇w) = e 0 , in Ω,…”

Section: Convergence Theoremmentioning

confidence: 99%

“…Weak Galerkin (WG) finite element method was first introduced by Ye and Wang for solving second order elliptic problems in [20]. Owing to its new features of flexibility in variational formulation and domain geometry, the WG method have been applied to various mathematical and engineering problems in many literatures, for instance, Stokes equation [21], Helmholtz equation [12], Maxwell equation [14], low regular elliptic problems [18], singularly perturbed convection-diffusion-reaction problems [9], and so on. In general, the standard differential operators are replaced by their weak forms, with optional stabilizer term to enforce weak continuity of the approximating functions.…”

mentioning

confidence: 99%

“…The latter has some interesting features including symmetric and positive definite systems, permission of arbitrary meshes, high-order convergence, and good performance on complicated interfaces. Recently, a class of relaxed weak Galerkin finite element methods has been presented for elliptic problems with low regularity in [18], by applying an over-relaxed factor to build up a connection with low regularity of solution and strengthen weak continuity across edges. In [10], Mu introduced a posteriori error estimate of weak Galerkin methods for the second-order elliptic interface problems, in which a promising algorithm was presented for adaptively solving interface problems.…”

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confidence: 99%

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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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Section: Convergence Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Song¹,

Qi²,

Liu³

et al. 2021

DCDS-B

Self Cite

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“…The weak Galerkin finite element method is an effective and flexible numerical technique for solving partial differential equations. The WG method was first introduced in [16] and then has been applied to solve various partial differential equations such as second order elliptic equations, biharmonic equations, Stokes equations, convection dominant problems, two-phase flow problems and Maxwell's equations [1,2,[4][5][6][7][8][10][11][12][13][14][17][18][19]. However, the standard a priori error analysis of weak Galerkin finite element methods requires additional regularity on solutions.…”

Section: Introductionmentioning

confidence: 99%

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Zhang¹

2021

NMTMA

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This paper presents error estimates in both an energy norm and the L 2norm for the weak Galerkin (WG) finite element methods for elliptic problems with low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for discontinuous finite element methods if the elliptic regularity is lower than H-1.5. Numerical results confirm the theoretical analysis.

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“…With these progress, the WG-FEM becomes more powerful and has been developed for solving many applications, such as elliptic interface problems, Stokes, Helmholtz, Maxwell etc [6,19,7,10,16]. Very recently, to overcome the difficulties on low regularity solutions, we also presented a relaxed WG method for solving elliptic and elliptic interface problems and analyzed L p error estimates [14,15].…”

mentioning

confidence: 99%

A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems

Liu¹,

Song²,

Zhao³

2021

DCDS-B

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In this article, we propose a new over-penalized weak Galerkin (OPWG) method with a stabilizer for second-order elliptic problems. 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 ] d), or (P k , P k−1 , [P k−1 ] d), with dimensions of space d = 2, 3. The method is absolutely stable with a constant penalty parameter, which is independent of mesh size and shaperegularity. We prove that for quasi-uniform triangulations, condition numbers of the stiffness matrices arising from the OPWG method are O(h −β 0 (d−1)−1), β 0 being the penalty exponent. Therefore we introduce a new mini-block diagonal preconditioner, which is proven to be theoretically and numerically effective in reducing the condition numbers of stiffness matrices to the magnitude of O(h −2). Optimal error estimates in a discrete H 1-norm and L 2-norm are established, from which the optimal penalty exponent can be easily chosen. Several numerical examples are presented to demonstrate flexibility, effectiveness and reliability of the new method.

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A Weak Galerkin Method with an Over-Relaxed Stabilization for Low Regularity Elliptic Problems

Cited by 6 publications

References 11 publications

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

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

Low Regularity Error Analysis for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems

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