Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources

Li, Hengguang; Wan, Xiang; Yin, Peimeng; Zhao, Lewei

doi:10.1016/j.cam.2021.113518

Cited by 12 publications

(8 citation statements)

References 14 publications

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“…We can see clearly that the error estimator guide the mesh refinements effectively around the points Q 1 and Q 2 . Here, Case 3 is an example in [31] solved by finite element method on graded meshes, which show optimal convergence rates with mesh refinements concentrating at the singular points Q 1 and Q 2 as well. The contours of the adaptive FEM approximations for different cases are shown in Figure 7 and Figure 8.…”

Section: 2mentioning

confidence: 99%

“…Here the vertices are Q 1 (0.2, 0.1), Q 2 (0.2, 0.9), Q 3 (0.7, 0.1) and Q 4 (0.9, 0.9). The domain and fractures were used in Example 5.2 in [31].…”

Section: 2mentioning

confidence: 99%

“…The transmission problem for the problem (1.1) is defined in the same domain of the original problem excluding on line fractures, and their solutions are exactly the same in the common region. The regularity of the solution excluding on line fractures of the original problem could also be obtained based on the equivalence to that of the transmission problem [31].…”

Section: Introductionmentioning

confidence: 99%

“…Finite element methods for the second order elliptic equations with singular source terms date back to 1970s, but the main focus was on point singular sources (see e.g., [6,36,37,12]). More recently, singular sources on complex geometry [21,22,31,23,14,13,3], including one dimensional (1D) fracture sources, have attracted more attention. (Adaptive) finite element methods were studied in [22,23] for problems involving a C 2 closed loop fracture strictly contained in the domain.…”

Section: Introductionmentioning

confidence: 99%

“…As a controlled equation in an optimal control problem, the boundary value problem (1.1) with a single C 2 curve fracture was solved in [21] by the linear finite element method. Later on, in order to improve the convergence rate of the finite element method in [21], Li et al in [31] investigated the problem (1.1) with one line segment fracture and the coefficient function g 1 = const, the regularities in both the whole domain and the region excluding line fractures were studied, and optimal finite element algorithm based on graded meshes was proposed. In this paper, we consider problem (1.1) with more general settings, namely, line fractures could be intersecting and disjoint, and the coefficient functions…”

Section: Introductionmentioning

confidence: 99%

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An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

Cao¹,

Li²,

Yi³

et al. 2021

Preprint

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In this paper, we study an adaptive finite element method for the elliptic equation with line Dirac delta functions as a source term.We investigate the regularity of the solution and the corresponding transmission problem to obtain the jump of normal derivative of the solution on line fractures. To handle the singularity of the solution, we adopt the meshes that conform to line fractures, and propose a novel a posteriori error estimator, in which the edge jump residual essentially use the jump of the normal derivative of the solution on line fractures. The error estimator is proven to be both reliable and efficient, finally an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests are presented to justify the theoretical findings.

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Section: 2mentioning

confidence: 99%

“…Here the vertices are Q 1 (0.2, 0.1), Q 2 (0.2, 0.9), Q 3 (0.7, 0.1) and Q 4 (0.9, 0.9). The domain and fractures were used in Example 5.2 in [31].…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

Cao¹,

Li²,

Yi³

et al. 2021

Preprint

Self Cite

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Anisotropic Error Estimates in Polyhedral Domains

2022

Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains

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Projection in negative norms and the regularization of rough linear functionals

et al. 2022

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In order to construct regularizations of continuous linear functionals acting on Sobolev spaces such as W 1,q 0 (Ω ), where 1 < q < ∞ and Ω is a Lipschitz domain, we propose a projection method in negative Sobolev spaces W −1,p (Ω ), p being the conjugate exponent satisfying p −1 + q −1 = 1. Our method is particularly useful when one is dealing with a rough (irregular) functional that is a member of W −1,p (Ω ), though not of L 1 (Ω ), but one strives for a regular approximation in L 1 (Ω ). We focus on projections onto discrete finite element spaces G n , and consider both discontinuous as well as continuous piecewise-polynomial approximations.While the proposed method aims to compute the best approximation as measured in the negative (dual) norm, for practical reasons, we will employ a computable, discrete dual norm that supremizes over a discrete subspace V m . We show that this idea leads to a fully discrete method given by a mixed problem on V m × G n . We propose a discontinuous as well as a continuous lowest-order pair, prove that they are compatible, and therefore obtain quasi-optimally convergent methods.We present numerical experiments that compute finite element approximations to Dirac delta's and line sources. We also present adaptively generate meshes, obtained from an error representation that comes with the method. Finally, we show how the presented projection method can be used to efficiently compute numerical approximations to partial differential equations with rough data.

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Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources

Cited by 12 publications

References 14 publications

An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

Anisotropic Error Estimates in Polyhedral Domains

Projection in negative norms and the regularization of rough linear functionals

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