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Gradient schemes for linear and non-linear elasticity equations

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Abstract

The gradient scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the gradient scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the gradient scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full \(H^2\)-regularity or for non-linear models.

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Notes

  1. The divergence of a tensor \({\varvec{\tau }}\) is taken row by row, i.e. if \({\varvec{\tau }}=({\varvec{\tau }}_{i,j})_{i,j=1,\ldots ,d}\) then \(\mathrm{div}({\varvec{\tau }})=(\sum _{j=1}^d\partial _j {\varvec{\tau }}_{i,j})_{i=1,\ldots ,d}\). This definition is consistent with our definition of \(\nabla \) by row: \(-\mathrm{div}\) is the formal dual operator of \(\nabla \).

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia

    Jérôme Droniou

  2. School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW, 2308, Australia

    Bishnu P. Lamichhane

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  1. Jérôme Droniou
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  2. Bishnu P. Lamichhane
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Correspondence to Jérôme Droniou.

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Droniou, J., Lamichhane, B.P. Gradient schemes for linear and non-linear elasticity equations. Numer. Math. 129, 251–277 (2015). https://doi.org/10.1007/s00211-014-0636-y

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