Related papers: A regularization approach for solving Poisson's eq…
A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…
The application of Renormalization Group (RG) methods to find perfect discretizations of partial differential equations is a promising but little investigated approach. We calculate the classically perfect fixed-point Laplace operator for…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
Bertaut's equivalent electric density idea (E. F. Bertaut, Journal de Physique {\bf 39}, 1331 (1978)) is applied to the case of two dimensional periodic continuous charge density distributions. The following derivation differs from what was…
The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the $L^2(\Omega)$-norm with order $1/2$ in convex…
We present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method…
A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across…
This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-\Delta u = f$ in $\mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's…
The ion distribution around charged colloids in solution has been investigated intensely during the last decade. However, few theoretical approaches have included the influence of variation in the dielectric permittivity within the system,…
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…
Electrical Impedance Tomography gives rise to the severely ill-posed Calder\'on problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing…
An inverse boundary value problem for the 1+1 dimensional wave equation $(\partial_t^2 - c(x)^2 \partial_x^2)u(x,t)=0,\quad x\in\mathbb{R}_+$ is considered. We give a discrete regularization strategy to recover wave speed $c(x)$ when we are…
We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that…
We consider a parameter identification problem related to a quasi-linear elliptic Neumann boundary value problem involving a parameter function $a(\cdot)$ and the solution $u(\cdot)$, where the problem is to identify $a(\cdot)$ on an…
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into…
We revisit the popular notion of effective or renormalized charge, which is a concept of central importance in the field of highly charged colloidal or polyelectrolyte solutions. Working at the level of a linear Debye-H\"uckel like theory…
For an ill-posed inverse problem, particularly with incomplete and limited measurement data, regularization is an essential tool for stabilizing the inverse problem. Among various forms of regularization, the lp penalty term provides a…
Complete and physically adequate analytical and semi-analytical solutions have been obtained using a practical dimensionless form of kinetic equation assuming azimuthal symmetry and Maxwellian distributions of target plasma species.…
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we…