We derive a compatible discretization method that relies heavily on the underlying geometric structure, and obeys the topological sequences and commuting properties that are constructed. As a sample problem we consider the vorticityvelocity-pressure formulation of the Stokes problem. We motivate the choice for a mixed variational formulation based on both geometric as well as physical arguments...

We study the space discretization of the Cauchy problem for a second order linear parabolic PDE, with one spacial 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 apply the modified variational iteration method MVIM for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution wi...

In this paper, we present a highly accurate Hamiltonian structure-preserving numerical method for simulating Hamiltonian wave equations. This method is obtained by applying spectral variational integrators (SVI) to the system of Hamiltonian ODEs which are derived from the spatial semi-discretization of the Hamiltonian PDE. The spatial variable is discretized by using high-order symmetric finite...

In this work, we introduce discontinuous Galerkin and enriched Galerkin formulations for the spatial discretization of phase-field fracture propagation. The nonlinear coupled system is formulated in terms of the Euler-Lagrange equations, which are subject to a crack irreversibility condition. The resulting variational inequality is solved in a quasi-monolithic way in which the irreversibility c...

The convexified Landau-Lifshitz minimization problem in micromagnetics leads to a degenerate variational problem. Therefore strong convergence of finite element approximations cannot be expected in general. This paper introduces a stabilized finite element discretization which allows for the strong convergence of the discrete magnetization fields with reduced convergence order for a uniaxial mo...

The price of an American style contract on assets driven by Lévy processes with infinite jump activity is expressed as solution of a parabolic variational integro-differential inequality (PIDI). A Galerkin discretization in logarithmic price using a wavelet basis is presented with compression of the moment matrix of the jump part of the price process’ Dynkin operator. An iterative solver with w...

We report results of a Monte Carlo simulation of the @4 quantum chain. In order to enhance the efficiency of the simulation we combine muItig~d simulation techniques with a refined discretization scheme. The resulting accuracy of our data allows for a significant test of an analytical approximation based on a va~ational ansatz. While the variational approximation is well reproduced for a large ...

This work is concerned with the numerical approximation of the unsteady Stokes flow of a viscous incompressible fluid driven by a threshold slip boundary condition of friction type. The continuous problem is formulated as variational inequality, which is next discretize in time based on backward Euler’s scheme. We prove existence and uniqueness of the solution of the time discrete problem by me...

In this paper, we study the a posteriori error estimate corresponding to Brinkman–Darcy–Forchheimer problem. We introduce variational formulation discretized by using finite element method. Then, establish an estimation with two types of indicators related discretization and linearization. Finally, numerical investigations are shown discussed.