Related papers: On Some Elliptic Interface Problems with Nonhomoge…
We consider parabolic equations with mixed boundary conditions and domain inhomogeneities supported on a lower dimensional hypersurface, enforcing a jump in the conormal derivative. Only minimal regularity assumptions on the domain and the…
The temperature and pressure jump coefficients at a liquid-vapor interface are calculated from the solution of the Shakhov kinetic model for the linearized Boltzmann equation. Complete and partial evaporation/condensation at the…
A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error…
In this paper, we are concerned with stable solutions , possibly unbounded and sign-changing, of some semi-linear elliptic problem with mixed nonlinear boundary conditions. We establish the nonexistence of stable solutions, the main methods…
We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally…
We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We…
In this paper, we consider the nonlocal elliptic problems in $\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we…
We introduce a new constructive method for establishing lower bounds on convergence rates of periodic homogenization problems associated with divergence type elliptic operators. The construction is applied in two settings. First, we show…
We use a diffuse interface method for solving Poisson's equation with a Dirichlet condition on an embedded curved interface. The resulting diffuse interface problem is identified as a standard Dirichlet problem on approximating regular…
Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…
We present an enhanced immersed interface method for simulating incompressible fluid flows in thin gaps between closely spaced immersed boundaries. This regime, common in engineered structures such as including tribological interfaces and…
We devise variants of classical nonconforming methods for symmetric elliptic problems. These variants differ from the original ones only by transforming discrete test functions into conforming functions before applying the load functional.…
Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales for heat transfer and elasticity. When microstructural features of a material have rapid, periodic fluctuations, the solution…
We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the Correction Function Method (CFM). The CFM offers a general framework to…
Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving…
Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.
In the present paper we consider linear and isotropic Maxwell equations with inhomogeneous interface conditions. We discretize the problem with the discontinuous Galerkin method in space and with the leapfrog scheme in time. An analytical…
This is a progress report on study of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review a…
We present modeling of the incompressible viscous flows in the domain containing an unconfined fluid and a porous medium. For such setting a rigorous derivation of the Beavers-Joseph-Saffman interface condition was undertaken by J\"ager and…
We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover…