Related papers: The finite volume method on Sierpi\'nski simplices
Numerous work revolve around the Sierpinski gasket. Its three-dimensional analogue, the Sierpinski tetrahedron, obtained by means of an iterative process which consists in repeatedly contracting a regular 3-simplex to one half of its…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
We construct a surface that is obtained from the octahedron by pushing out 4 of the faces so that the curvature is supported in a copy of the Sierpinski gasket in each of them, and is essentially the self similar measure on SG. We then…
We define and study fractional stable random fields on the Sierpi\'nski gasket. Such fields are formally defined as $(-\Delta)^{-s} W_{K,\alpha}$, where $\Delta$ is the Laplace operator on the gasket and $W_{K,\alpha}$ is a stable random…
We propose a finite volume stochastic collocation method for the random Euler system. We rigorously prove the convergence of random finite volume solutions under the assumption that the discrete differential quotients remain bounded in…
The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are…
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and the 3-dimensional Sierpinski gasket, but the method is expected to be effective for…
We propose a stochastic collocation method based on the piecewise constant interpolation on the probability space combined with a finite volume method to solve the compressible Navier-Stokes system at the nodal points. We show convergence…
A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of…
In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpi\'nski gasket in $\mathbb{R}^2$. Parameters $\lambda, \gamma, \alpha, \beta$ are real and $1<q<p<\alpha+\beta.$ Functions $a,b,h :…
We prove the hydrodynamic limit for Glauber-Kawasaki dynamics on the Sierpi\'nski gasket, a prototypical fractal graph that lacks translational invariance. The main novelty lies in incorporating Glauber dynamics, allowing for particle…
We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not…
We study nonlinear hyperbolic conservation laws posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and defined from a prescribed flux field of n-forms depending on a parameter (the unknown variable), a class of…
By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous…
We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency…
In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show…
This paper is concerned with the quantitative homogenization of the steady Stokes equations with the Dirichlet condition in a periodically perforated domain. Using a compactness method, we establish the large-scale interior $C^{1, \alpha}$…
In this paper, we study discrete approximations of semi-Dirichlet forms obtained by adding non-symmetric drift terms, expressed in terms of mutual energy measures, to resistance forms whose associated resistance metric spaces are compact.…
We build convergent discretizations and semi-implicit solvers for the Infinity Laplacian and the game theoretical $p$-Laplacian. The discretizations simplify and generalize earlier ones. We prove convergence of the solution of the Wide…
We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The…