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We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the…
In this paper we study the asymptotic behavior of solutions to systems of strongly coupled integral equations with oscillatory coefficients. The system of equations is motivated by a peridynamic model of the deformation of heterogeneous…
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study…
The paper presents a new reduction method designed for dynamic contact problems. Recently, we have proposed an efficient reduction scheme for the node-to-node formulation, that leads to Linear Complementarity Problems (LCP). Here, we…
The overreaching goal of this paper is to investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological…
Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of…
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…
In this paper we will consider the peridynamic equation of motion which is described by a second order in time partial integro-differential equation. This equation has recently received great attention in several fields of Engineering…
We derive an efficient algorithm to find solutions to Euler's concordant form problem and rational points on elliptic curves associated with this problem.
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the…
Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally H\"older continuous, provided coefficients are…
We prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed point index. Some of the…
We study the removability of a singular set in the boundary of Neumann problem for elliptic equations with variable exponent. We consider the case where the singular set is compact, and give sufficient conditions for removability of this…
In this work we establish existence and multiplicity of solutions for elliptic problem with nonlinear boundary conditions under strong resonance conditions at infinity. The nonlinearity is resonance at infinity and the reso- nance phenomena…
We use novel integral representations developed by the second author to prove certain rigorous results concerning elliptic boundary value problems in convex polygons. Central to this approach is the so-called global relation, which is a…
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 consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…
Numerical and analytical methods are developed for the investigation of contact sets in electrostatic-elastic deflections modeling micro-electro mechanical systems. The model for the membrane deflection is a fourth-order semi-linear partial…
Almost all materials are anisotropic. In this paper, interface relations of anisotropic elliptic partial differential equations involving discontinuities across interfaces are derived in two and three dimensions. Compared with isotropic…