Related papers: Generalized Gaffney inequality and discrete compac…
We consider nonhomogeneous fractional $p$-Laplace equations defined on a bounded nonsmooth domain which goes beyond the Lipschitz category. Under a sufficient flatness assumption on the domain in the sense of Reifenberg, we establish…
A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\Omega$ is a general smooth domain with a curved…
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…
We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is…
This note constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding…
We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish…
We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the…
We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass…
This paper proposes a $C^{0}$ (non-Lagrange) primal finite element approximation of the linear elliptic equations in non-divergence form with oblique boundary conditions in planar, curved domains. As an extension of [Calcolo, 58 (2022), No.…
In this paper we develop numerical analysis for finite element discretization of semilinear elliptic equations with potentially non-Lipschitz nonlinearites. The nonlinearity is essecially assumed to be continuous and monotonically…
Given a compact metric graph $\Gamma$ and the Laplacian $\Delta_{\Gamma}$ coupled with standard (Kirchhoff) vertex conditions, solutions to fractional elliptic partial differential equations of the form $(\kappa^2 -…
In this preprint we consider generalizations of discrete and integral Cauchy--Bunyakovskii inequalities by the method of mean values with some applications. Mostly the material is compiled as a short survey but some results are proved. Main…
We investigate some analytic properties of traces of Dirichlet forms with respect to measures satisfying Hardy-type inequality. Among other results we prove convergence of spectra, ordered eigenvalues, eigenfunctions as well as convergence…
This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary…
Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
In this PhD thesis we introduce a generalized fractional calculus of variations. We consider variational problems containing generalized fractional integrals and derivatives, and study them using standard (indirect) and direct methods. In…
We present a weak finite element method for elliptic problems in one space dimension. Our analysis shows that this method has more advantages than the known weak Galerkin method proposed for multi-dimensional problems, for example, it has…