We consider in this paper numerical approximations of two-phase incompressible flows with different densities and viscosities. We present a variational derivation for a thermodynamically consistent phase-field model that admits an energy law. Two decoupled time discretization schemes for the coupled nonlinear phase-field model are constructed and shown to be energy stable. Numerical experiments...

We study the space discretization of the Cauchy problem for a second order linear parabolic PDE, with one spatial dimension and unbounded time and space-dependent coefficients. The equation free term and the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach, and approximate in space the PDE probl...

We propose a First-Order System Least Squares (FOSLS) method based on deep-learning for numerically solving second-order elliptic PDEs. The we is capable of dealing with either variational and non-variational problems, because its meshless nature, it can also deal problems posed in high-dimensional domains. prove the Γ-convergence neural network approximation towards solution continuous problem...

We consider the Navier-Stokes-Alpha model as an approximation of turbulent flows under realistic, non-periodic, boundary conditions. We derive that the variational formulation of Navier-Stokes-Alpha model under non-periodic boundary conditions, and prove that it has a unique weak solution. Next we consider finite element approximation of the model. We give semi discretization of the model and p...

Keywords hp-FEM · contact problems This paper presents mixed variational formulation and its discretization with finite elements of higher-order for Signorini's problem with Tresca's friction. To guarantee the unique existence of the discrete saddle point of the mixed method, a discrete inf-sup condition is proven. Moreover, a solution scheme based on the dual formulation of the mixed method is...

In this paper, we study the adaptive mixed finite element methods and variational discretization for optimal control problems governed by nonlinear elliptic equations. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discretized. Then we derive a posteriori error estimates both for the coupled state and the control ...

In the present article, we apply the modified piecewise variational iteration method to obtain the approximate analytical solutions of the differential equations with piecewise continuous arguments. This technique provides a sequence of functions which converges to the exact solution of the problem. Moreover, this method reduces the volume of calculations because it does not need discretization...

Questions of solution of three-dimensional diffraction problems are considered. Each problem is formulated as a single weakly-singular integral equation of the 1st kind for a single unknown function. Discretization of these equations is realized by means of special smoothing method of integral operators kernels. Numerical solutions of systems of linear algebraic equations, approximating integra...

In this paper we consider an optimal control problem governed by a time-dependent variational inequality arising in quasistatic plasticity with linear kinematic hardening. We address certain continuity properties of the forward operator, which imply the existence of an optimal control. Moreover, a discretization in time is derived and we show that every local minimizer of the continuous problem...

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup di...