A priori 𝐿^{𝜌} error estimates for Galerkin approximations to porous medium and fast diffusion equations

An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two-dimensional numerical results are presented.

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“…by Wei and Lefton [56], where d is the space dimension. It is remarked that these estimates are obtained for quasi-uniform meshes.…”

Section: Introductionmentioning

confidence: 99%

A study on moving mesh finite element solution of the porous medium equation

Ngo

Huang

2017

Journal of Computational Physics

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“…where m(v) is a maximal monotone graph in R×R possibly with a singularity at the origin [m (0) = ∞]. In their numerical analysis they use a smoothing procedure that regularizes m. Other relevant numerical studies using some sort of regularization techniques in the approximation of solutions of degenerate parabolic equations include [4], [23], [25] and [30]. Of particular interest is the work presented in [26], which may be thought of as the numerical analysis counterpart of the current paper.…”

Section: Methodsmentioning

confidence: 99%

On the diffusive wave approximation of the shallow water equations

2008

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In this paper, we study basic properties of the diffusive wave approximation of the shallow water equations (DSW). This equation is a doubly non-linear diffusion equation arising in shallow water flow models. It has been used as a model to simulate water flow driven mainly by gravitational forces and dominated by shear stress, that is, under uniform and fully developed turbulent flow conditions. The aim of this work is to present a survey of relevant results coming from the studies of doubly non-linear diffusion equations that can be applied to the DSW equation when topographic effects are ignored. In fact, we present proofs of the most relevant results existing in the literature using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions.

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“…A sketch of the proof for α = β is given in Wei and Lefton [63]. An interested reader is referred to [63], [58], and [27] for further details.…”

Section: Then We Have ||V − πV|| Mβ Chmentioning

confidence: 99%

“…An interested reader is referred to [63], [58], and [27] for further details. For each ε > 0 and f ∈ W −1,r ′ (Ω), we consider u ε h ∈ S h 0 (Ω) to be the unique solution of…”

Section: Then We Have ||V − πV|| Mβ Chmentioning

confidence: 99%