Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions

Lin, Runchang

doi:10.1137/070700267

Cited by 33 publications

(19 citation statements)

References 28 publications

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“…This behavior is caused by a small interval of width O(ǫ) or O( √ ǫ) (so-called boundary layer) in which u ′′ rapidly changes. Since the solution of the standard FEM exhibits large nonphysical oscillation, many stabilization techniques have been suggested, including upwind, streamline-diffusion FEMs, discontinuous Galerkin methods, and least-squares FEMs [13,14,17,20]. Another effective way for globally solving this problem is to construct a numerical scheme on a layer adapted mesh, such as Shishkin type meshes and Bakhvalov type meshes.…”

Section: −ǫUmentioning

confidence: 99%

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

Zhu

Tian

Zhang

2011

Communications in Mathematical Sciences

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Abstract. In this paper the local discontinuous Galerkin method (LDG) is considered for solving one-dimensional singularly perturbed two-point boundary value problems of convection-diffusion type and reaction-diffusion type. Error estimates are studied on Shishkin meshes. The L 2 error bounds for the LDG approximation of the solution and its derivative are uniformly valid with respect to the singular perturbation parameter. Numerical experiments indicate that the orders of convergence are sharp.Key words. Local discontinuous Galerkin method, singularly perturbed, Shishkin mesh.AMS subject classifications. 65L11, 65N30, 65N15. IntroductionThe discontinuous Galerkin (DG) method was introduced in 1973 as a way of solving the steady-state neutron transport equation [16]. Successively in 1974, Lesaint and Raviart made the first analysis for the linear advection equation [12]. Since then many DG methods were vigorously studied. Among these is the local discontinuous Galerkin (LDG) method, which was proposed in [3] and [9] by separating higher order operators into systems of first order equations so that classical DG methods can be extended to problems with second order operators, especially for convection-diffusion and hyperbolic equations. The state of the art of the development of these methods and their applications can be found in [1,2,5,7,10].In this work, we apply the LDG method to 1-D singularly perturbed problems with Dirichlet boundary conditions:

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Section: −ǫUmentioning

confidence: 99%

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

Zhu

Tian

Zhang

2011

Communications in Mathematical Sciences

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“…Our experiments indicate that, when λ i and µ i are properly selected, the method has better computational performance, though the convergence rate remains the same; cf. [35].…”

Section: Remarkmentioning

confidence: 99%

“…Nevertheless, the DGLSFEM proposed here is very robust with respect to the numerical fluxes defined in (19) and (20). [35] are used in the proof of Theorem 1, which nonetheless have been adjusted to treat difficulties caused by the interfaces. In particular, continuous shape functions are required for nodes on Γ 0 , as indicated in (8).…”

Section: Remarkmentioning

confidence: 99%

“…Recently, a discontinuous Galerkin least-squares finite element method (DGLS-FEM) has been developed for singularly perturbed reaction-diffusion problems with smooth data and regular boundary [35]. The method is stable and robust, which, unlike the standard DG method, does not involve user-chosen problem-dependent parameters.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we extend the method proposed in [35] and develop an effective methodology for singularly perturbed problems with discontinuous coefficients and boundary singularities. In Sect.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

Lin

2009

Numer. Math.

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In this paper, a discontinuous Galerkin least-squares finite element method is developed for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. In a companion paper (Lin in SIAM J Numer Anal 47:89-108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.

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Least‐squares virtual element method for the convection‐diffusion‐reaction problem

Wang

2021

Numerical Meth Engineering

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In this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the H(div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

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Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions

Cited by 33 publications

References 28 publications

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

Convergence analysis of the LDG method for singularly perturbed two-point boundary value problems

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

Least‐squares virtual element method for the convection‐diffusion‐reaction problem

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