Keywords: Nonlinear integro-differential equations Finite elements Galerkin method a b s t r a c t Galerkin finite element method for the approximation of a nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. First type initial-boundary value problem is investigated. The convergence of the finite element scheme is proved. The ...

This paper presents an effective three-dimensional (3D) nonlinear explicit dynamic meshfree algorithm for the simulation of soft tissue mechanical responses. In the algorithm soft tissues are considered to be hyperelastic and nearly incompressible materials. The algorithm is based on the element-free Galerkin (EFG) method using total Lagrangian formulation and moving least square (MLS) approxim...

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in L∞(L2). These estimates are valid under weak restrictions on the space-time mesh.

This paper deals with the finite element solution of the convection diffusion equation in one and two dimensions. Two main techniques are adopted and compared. The first one includes Petrov-Galerkin based on Lagrangian tensor product elements in conjunction with streamlined upwinding. The second approach represents Bubnov/Petrov-Galerkin schemes based on a new group of exponential elements. It ...

For nearly incompressible elasticity, volumetric locking is a well-known phenomenon with low-order (cubic or lower) fi nite element method methods, of which continuous extended fi nite element methods (XFEMs) are no exception. We will present an XFEM that is simultaneously lock-free and optimally convergent. Based on our earlier work of an optimally convergent discontinuous-Galerkin-based XFEM,...

In order to solve the elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured meshes, a nodal Discontinuous Galerkin Finite Element Method (DG-FEM) is presented, which combines the geometrical flexibility of the Finite Element Method and strongly nonlinear wave simulation capability of the Finite Volume Method. The equations of nonlinear elastody...

We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive...

We prove in an abstract setting that standard (continuous) Galerkin finite element approximations are the limit of interior penalty discontinuous Galerkin approximations as the penalty parameter tends to infinity. We apply this result to equations of non-negative characteristic form and the non-linear, time dependent system of incompressible miscible displacement. Moreover, we investigate varyi...