132 research outputs found
A Moving Mesh Method for Porous Medium Equation by the Onsager Variational Principle
In this paper, we introduce a new approach to solving the porous medium
equation using a moving mesh finite element method that leverages the Onsager
variational principle as an approximation tool. Both the continuous and
discrete problems are formulated based on the Onsager principle. The energy
dissipation structure is maintained in the semi-discrete and fully implicit
discrete schemes. We also develop a fully decoupled explicit scheme by which
only a few linear equations are solved sequentially in each time step. The
numerical schemes exhibit an optimal convergence rate when the initial mesh is
appropriately selected to ensure accurate approximation of the initial data.
Furthermore, the method naturally captures the waiting time phenomena without
requiring any manual intervention
An Eulerian space-time finite element method for diffusion problems on evolving surfaces
In this paper, we study numerical methods for the solution of partial
differential equations on evolving surfaces. The evolving hypersurface in
defines a -dimensional space-time manifold in the space-time
continuum . We derive and analyze a variational formulation for
a class of diffusion problems on the space-time manifold. For this variational
formulation new well-posedness and stability results are derived. The analysis
is based on an inf-sup condition and involves some natural, but non-standard,
(anisotropic) function spaces. Based on this formulation a discrete in time
variational formulation is introduced that is very suitable as a starting point
for a discontinuous Galerkin (DG) space-time finite element discretization.
This DG space-time method is explained and results of numerical experiments are
presented that illustrate its properties.Comment: 22 pages, 5 figure
Γ-convergence Approximation of Fracture and Cavitation in Nonlinear Elasticity
The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0820-3Our starting point is a variational model in nonlinear elasticity that allows for cavitation and fracture that was introduced by Henao and Mora-Corral (Arch Rational Mech Anal 197:619–655, 2010). The total energy to minimize is the sum of the elastic energy plus the energy produced by crack and surface formation. It is a free discontinuity problem, since the crack set and the set of new surface are unknowns of the problem. The expression of the functional involves a volume integral and two surface integrals, and this fact makes the problem numerically intractable. In this paper we propose an approximation (in the sense of Γ-convergence) by functionals involving only volume integrals, which makes a numerical approximation by finite elements feasible. This approximation has some similarities to the Modica–Mortola approximation of the perimeter and the Ambrosio–Tortorelli approximation of the Mumford–Shah functional, but with the added difficulties typical of nonlinear elasticity, in which the deformation is assumed to be one-to-one and orientation-preservingD. Henao gratefully acknowledges the Chilean Ministry of Education’s support through the FONDE-CYT Iniciación project no. 11110011. C. Mora-Corral has been supported by Project MTM2011-28198 of the Spanish Ministry of Economy and Competitivity, the ERC Starting grant no. 307179, the “Ramón y Cajal” programme and the European Social Fund. X. Xu acknowledges the funding by NSFC 1100126
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