The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Zhang, Tie Zhu; Zhang, Shu Hua

doi:10.1007/s10492-015-0112-8

Cited by 1 publication

(1 citation statement)

References 29 publications

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“…To our best knowledge, superconvergence on two levels and superconvergence from perturbed data have never been mentioned in literature. Several studies have addressed the one order of superconvergence, especially for finite element [40,23,2] and finite volume type methods [8,3,38]. Regarding finite difference methods on polygonal domains, Ferreira and Grigorieff have shown in [13] one level of superconvergence at the second order of accuracy for general elliptic operators while Li et al in [26,24,25] specifically studied the Shortley-Weller scheme.…”

Section: Superconvergence: State Of the Artmentioning

confidence: 99%

Superconvergent second order Cartesian method for solving free boundary problem for invadopodia formation

Gallinato

Poignard

2017

Journal of Computational Physics

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In this paper, we present a superconvergent second order Cartesian method to solve a free boundary problem with two harmonic phases coupled through the moving interface. The model recently proposed by the authors and colleagues describes the formation of cell protrusions. The moving interface is described by a level set function and is advected at the velocity given by the gradient of the inner phase. The finite differences method proposed in this paper consists of a new stabilized ghost fluid method and second order discretizations for the Laplace operator with the boundary conditions (Dirichlet, Neumann or Robin conditions). Interestingly, the method to solve the harmonic subproblems is superconvergent on two levels, in the sense that the first and second order derivatives of the numerical solutions are obtained with the second order of accuracy, similarly to the solution itself. We exhibit numerical criteria on the data accuracy to get such properties and numerical simulations corroborate these criteria. In addition to these properties, we propose an appropriate extension of the velocity of the level-set to avoid any loss of consistency, and to obtain the second order of accuracy of the complete free boundary problem. Interestingly, we highlight the transmission of the superconvergent properties for the static subproblems and their preservation by the dynamical scheme. Our method is also well suited for quasistatic Hele-Shaw-like or Muskat-like problems.

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