Uniformly stable semidiscrete approximation schemes are desirable for constructing approximate solutions of certain optimal control problems for distributed parameter systems. We discuss construction of uniformly stable spline-based approximation schemes for a scalar delay equation. A method is used which employs an equivalent inner product. Convergence is established via a `hybrid' Trotter-Kat...

We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.

An approach to calculating approximate solutions to the continuum Schwinger– Dyson equations is outlined, with examples for φ 4 in D = 1. This approach is based on the source Galerkin methods developed by García, Guralnik and Law-son. Numerical issues and opportunities for future calculations are also discussed briefly.

The present investigation analyses the critical buckling studies of triple-walled carbon nanotube using Euler─Bernoulli model. study deals with three different boundary conditions, namely, simply-simply, clamped-clamped, and clamped-simply supported nanotube. Using Bubnov─Galerkin Petrov─Galerkin methods, continuum model estimates load. main advantage these two approximate methods is to obtain ...

This paper furnishes two classes of methods for calculating the best approximate solution of an operator equation in Banach spaces, where the operator is bounded, linear and has closed range. The best approximate solution can be calculated by an iterative method in Banach spaces stated in terms of an operator parameter. Specifying the parameter yields some new and some old iterative techniques....

We numerically verify that the non-symmetric interior penalty Galerkin method and the Oden-Babus̆ka-Baumann method have sub-optimal convergence properties when measured in the L-norm for odd polynomial approximations. We provide numerical examples that use piece-wise linear and cubic polynomials to approximate a second-order elliptic problem in one and two dimensions.

We consider a model of population dynamics whose mortality function is unbounded. We approximate the solution of the model using a discontinuous Galerkin finite element for the age variable and a backward Euler for the time variable. We present several numerical examples. It is experimentally shown that the scheme converges at the rate of h3/2 in the case of piecewise linear polynomial space.

Based on H-Galerkin mixed finite element method with nonconforming quasi-Wilson element, a numerical approximate scheme is established for pseudo-hyperbolic equations under arbitrary quadrilateral meshes. The corresponding optimal order error estimate is derived by the interpolation technique instead of the generalized elliptic projection which is necessary for classical error estimates of fini...

The standard in rod finite element formulations is the Bubnov–Galerkin projection method, where test functions arise from a consistent variation of ansatz functions. This approach becomes increasingly complex when highly nonlinear are chosen to approximate rod's centerline and cross-section orientations. Using Petrov–Galerkin we propose whole family nodal generalized virtual displacements veloc...

The use of " ideal " optimal test functions in a Petrov-Galerkin scheme guarantees the discrete stability of the variational problem. However, in practice, the computation of the ideal optimal test functions is computationally intractable. In this paper, we study the effect of using approximate, " practical " test functions on the stability of the DPG (discontinuous Petrov-Galerkin) method and ...