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Abstract In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.
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Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective
In the mean field integrate-and-fire model, the dynamics of a typical neuronwithin a large network is modeled as a diffusion-jump stochastic process whosejump takes place once the voltage reaches a threshold. In this work, the maingoal is to establish the convergence relationship between the regularizedprocess and the original one where in the regularized process, the jumpmechanism is replaced by a Poisson dynamic, and jump intensity within theclassically forbidden domain goes to infinity as the regularization parametervanishes. On the macroscopic level, the Fokker-Planck equation for the processwith random discharges (i.e. Poisson jumps) are defined on the whole space,while the equation for the limit process is on the half space. However, withthe iteration scheme, the difficulty due to the domain differences has beengreatly mitigated and the convergence for the stochastic process and the firingrates can be established. Moreover, we find a polynomial-order convergence forthe distribution by a re-normalization argument in probability theory. Finally,by numerical experiments, we quantitatively explore the rate and the asymptoticbehavior of the convergence for both linear and nonlinear models.
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- PAR ID:
- 10355164
- Date Published:
- Journal Name:
- Mathematical Neuroscience and Applications
- Volume:
- Volume 1
- ISSN:
- 2801-0159
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46298/mna.7203
