Related papers: On regularizations of the delta distribution
Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this…
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the…
Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic…
In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\, \Omega,\newline…
The Lax equivalence theorem guarantees convergence of stable and consistent discretizations for linear hyperbolic partial differential equations (PDEs). For nonlinear problems, however, stability and consistency alone do not generally…
Higher-order regularization problem formulations are popular frameworks used in machine learning, inverse problems and image/signal processing. In this paper, we consider the computational problem of finding the minimizer of the Sobolev…
In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one…
A smoothness-increasing accuracy conserving filtering approach to the regularization of discontinuities is presented for single domain spectral collocation approximations of hyperbolic conservation laws. The filter is based on convolution…
The present study investigates a linear-quadratic Dirichlet control problem governed by a non-coercive elliptic equation posed on a possibly non-convex polygonal domain. Tikhonov regularization is carried out in an energy seminorm. The…
We explore how the analysis of the Carleman linearization can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…
In this paper we complement the program concerning the application of symmetrization methods to nonlocal PDEs by providing new estimates, in the sense of mass concentration comparison, for solutions to linear fractional elliptic and…
We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally…
This work presents a numerical study of the Dirichlet problem for the fractional Laplacian $(-\Delta)^s$ with $s\in(0,1)$ using Finite Element methods with non-standard bases. Classical approaches based on piece-wise linear basis yield…
It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent…
We investigate linear parabolic equations in divergence form with singular coefficients and non-smooth boundary data. When the diffusion, drift, or potential terms, as well as the initial or boundary conditions, are distributions rather…
We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…
We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic…
We consider the electromagnetic field in the presence of polarizable point dipoles. In the corresponding effective Maxwell equation these dipoles are described by three dimensional delta function potentials. We review the approaches…
In this paper, we examine the regularity of the solutions to the double-divergence equation. We establish improved H\"older continuity as solutions approach their zero level-sets. In fact, we prove that $\alpha$-H\"older continuous…