Related papers: Solving parabolic equations on the unit sphere via…
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution…
In this paper, we consider the time change of the diffusion process on the 2-dimensional unit sphere generated by the Laplace-Beltrami operator, perturbed by means of a longitudinal vector field. First, this is done by addressing the…
We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the…
We present a spectral method for parabolic partial differential equations with zero Dirichlet boundary conditions. The region {\Omega} for the problem is assumed to be simply-connected and bounded, and its boundary is assumed to be a smooth…
We develop a method to obtain the general solution of the Laplace equation in $d$-dimension in ultraspherical coordinates.
We propose a least squares formulation for abstract parabolic equations in the natural $L^2(0,T;V^\star)\times H$ norm which only relies on natural regularity assumptions on the data of the problem. The resulting bilinear form then is…
In this work, the propagation of an ultrasonic pulse in a thin plate is computed solving the differential equations modeling this problem. To solve these equations finite differences are used to discretize the temporal variable, while…
We present and analyze a multiscale method for wave propagation problems, posed on spatial networks. By introducing a coarse scale, using a finite element space interpolated onto the network, we construct a discrete multiscale space using…
We study elliptic and parabolic problems governed by the singular elliptic operators \begin{equation*} \mathcal L =y^{\alpha_1}\Delta_{x} +y^{\alpha_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right), \qquad\alpha_1, \alpha_2 \in\mathbb R…
This paper presents a mixed basis approach for Laplace eigenvalue problems, which treats the boundary as a perturbation of the free Laplace operator. The method separates the boundary from the volume via a generic function that can be…
We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and…
We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations…
We study the numerical approximation of linear-quadratic optimal control problems subject to the fractional Laplace equation with its spectral definition. We compute an approximation of the state equation using a discretization of the…
We study the stochastic $p$-Laplace system in a bounded domain. We propose two new space-time discretizations based on the approximation of time-averaged values. We establish linear convergence in space and $1/2$ convergence in time.…
We present algorithms for solving spatially nonlocal diffusion models on the unit sphere with spectral accuracy in space. Our algorithms are based on the diagonalizability of nonlocal diffusion operators in the basis of spherical harmonics,…
The purpose of this article is to construct highly localized summability kernels on the unit sphere in ${\mathbb R}^d$ that are restrictions to the sphere of linear combinations of a small number of shifts of the fundamental solution of the…
In this paper, we present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres…
We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on…