A Lumped Mass Finite-element Method with Quadrature for a Non-linear Parabolic Problem

Nie, Yaguang; Thomée, Vidar

doi:10.1093/imanum/5.4.371

Cited by 46 publications

(35 citation statements)

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“…where u H ∈ S 0 (Ω, T H ) is the output of the online algorithm 3.2 and the functionsψ m,s N,x0 ∈ S N (Y ) can be evaluated using the RB decomposition (22) with the RB produced by the Greedy algorithm 2.3. Notice that the coefficients α (x,s,m) j in (22) are already computed (24) and need not to be recomputed.…”

Section: Remark 31mentioning

confidence: 99%

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An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

Abdulle

Bai

Vilmart

2014

Numerical Meth Engineering

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SUMMARYInspired by recent analyses of the finite element heterogeneous multiscale method and the reduced basis technique for nonlinear problems, we present a simple and concise finite element algorithm for the reliable and efficient resolution of elliptic or parabolic multiscale problems of nonmonotone type. Solutions of appropriate cell problems on sampling domains are selected by a greedy algorithm in an offline stage and assembled in a reduced basis (RB). This RB is then used in an online stage to solve two-scale problems at a computational cost comparable to the single-scale case. Both the offline and the online cost are independent of the smallest scale in the physical problem. The performance and accuracy of the algorithm are illustrated on 2D and 3D stationary and evolutionary nonlinear multiscale problems.KEY WORDS: nonlinear nonmonotone problems, numerical homogenization, reduced basis method, a posteriori error estimator, finite element method

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Section: Remark 31mentioning

confidence: 99%

“…Notice that the coefficients α (x,s,m) j in (22) are already computed (24) and need not to be recomputed.…”

Section: Remark 31mentioning

confidence: 99%

An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

Abdulle

Bai

Vilmart

2014

Numerical Meth Engineering

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“…The solution φ of (3.3) is also generally second order accurate in 2D [35]. Let D denote a block on the diagonal ofD.…”

Section: Relation To Microscopic Equationsmentioning

confidence: 99%

Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes

Engblom¹,

Ferm²,

Hellander³

et al. 2009

SIAM J. Sci. Comput.

120

203

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Stochastic chemical systems with diffusion are modeled with a reactiondiffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reactiondiffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso-or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.

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“…For example, Thomeé [8], Chen and Huang [1] studied superconvergence of the gradient in L 2 norm while Thomeé et al [9] studied maximum norm superconvergence of gradient for linear finite element. Superconvergence of the lumped finite element method for linear and nonlinear parabolic problems were studied in [2] and [6], respectively. In this paper, we introduce a different way of approximating the initial condition, namely (1.4) and investigate the superconvergence of finite element for parabolic problem using any order element.…”

Section: We Consider the Following Map U H (T) : [0t ] → S H Definedmentioning

confidence: 99%

Superconvergence of finite element method for parabolic problem

Kwak

Lee

2000

International Journal of Mathematics and Mathematical Sciences

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A Lumped Mass Finite-element Method with Quadrature for a Non-linear Parabolic Problem

Cited by 46 publications

References 0 publications

An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

An offline–online homogenization strategy to solve quasilinear two‐scale problems at the cost of one‐scale problems

Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes

Superconvergence of finite element method for parabolic problem

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