Related papers: Nitsche's method for a Robin boundary value proble…
This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical…
In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous…
We consider the reconstruction of a heterogeneous coefficient field in a Robin boundary condition on an inaccessible part of the boundary in a Poisson problem with an uncertain (or unknown) inhomogeneous conductivity field in the interior…
We compare different training strategies for the Deep Ritz Method for elliptic equations with Dirichlet boundary conditions and highlight the problems arising from the boundary values. We distinguish between an exact resolution of the…
We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are…
In this paper we study the $p$-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems…
The Poisson--Boltzmann equation is widely used to model electrostatics in molecular systems. Available software packages solve it using finite difference, finite element, and boundary element methods, where the latter is attractive due to…
This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schr\"odinger equations posed in a half plane $\mathbb{R} \times \mathbb{R}^+$ with nonhomogeneous Dirichlet boundary conditions. For any given $s \ge 0$, if the…
In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly…
The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in…
We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The…
We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter…
In the present work, we discuss variational regularization for ill-posed nonlinear problems with focus on an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation…
We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains…
We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the…
This paper investigates simple bilevel optimization problems where we minimize an upper-level objective over the optimal solution set of a convex lower-level objective. Existing methods for such problems either only guarantee asymptotic…
We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and…
We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is…
We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem…
Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce…