Related papers: Finite Element Convergence for the Joule Heating P…
Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an $L^{2}$-setting of monomially weighted spaces. A mathematical framework is developed which…
We present a numerical analysis of a higher order unfitted space-time Finite Element method applied to a convection-diffusion model problem posed on a moving bulk domain. The method uses isoparametric space-time mappings for the geometry…
Higher regularity estimate has been a challenging question for the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in [15] even for symmetric convex…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
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…
The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients…
This paper studies a two-material optimal design problem for the time-averaged duality pairing between a (possibly time-dependent) heat source and the weak solution of an initial-boundary value problem for the heat equation with a…
In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method and studied for Poisson problem in [Renard]. The method allows computations in domains whose boundaries…
Solutions to the wave equation in the exterior of a polyhedral domain or a screen in $\mathbb{R}^3$ exhibit singular behavior from the edges and corners. We present quasi-optimal $hp$-explicit estimates for the approximation of the…
It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem…
We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains…
We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for…
In this paper, we examine the effectiveness of classic multiscale finite element method (MsFEM) (Hou and Wu, 1997; Hou et al., 1999) for mixed Dirichlet-Neumann, Robin and hemivariational inequality boundary problems. Constructing so-called…
We discuss the initial boundary value problem for a heat equation in a domain surrounded by a layer. The main features of this problem are twofold: on one hand, the layer is thin compared to the scale of the domain, and on the other hand,…
This paper introduces a nonconforming virtual element method for general second-order elliptic problems with variable coefficients on domains with curved boundaries and curved internal interfaces. We prove arbitrary order optimal…
We study the Boltzmann equation in a smooth bounded domain featuring a mixed boundary condition. Specifically, gas particles experience specular reflection in two parallel plates, while diffusive reflection occurs in the remaining portion…
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We…
This article investigates adaptive mesh refinement procedures for the time-domain wave equation with Neumann boundary conditions, formulated as an equivalent hypersingular boundary integral equation. Space-adaptive and time-adaptive…
With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…