In this article, element free Galerkin method is used for static analysis of thin micro/nanoscale plates based on the nonlocal plate theory. The problem is solved for the plates with arbitrary boundary conditions. Since shape functions of the element free Galerkin method do not satisfy the Kronecker’s delta property, the penalty method is used to impose the essential boundary conditions. Discre...

We present in this paper several extremely efficient and accurate spectral-Galerkin methods for secondand fourth-order equations in polar and cylindrical geometries. These methods are based on appropriate variational formulations which incorporate naturally the pole condition(s). In particular, the computational complexities of the Chebyshev–Galerkin method in a disk and the Chebyshev–Legendre–...

We introduce a new and efficient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the LegendreGalerkin and Chebyshev-Galerkin methods.

In this article we present a multilevel preconditioner for interior penalty discontinuous Galerkin discretizations of second order elliptic boundary value problems that gives rise to uniformly bounded condition numbers without any additional regularity assumptions on the solution. The underlying triangulations are only assumed to be shape regular but may have hanging nodes subject to certain mi...

We discuss the methodology of the validation of a higher order discontinuous Galerkin scheme for acoustic computations. That includes an accurate definition of the exact solution in the problem as well as careful study of convergence properties of a DG scheme for a chosen acoustic problem. The efficiency of a higher order scheme will be confirmed for computations on coarse meshes.

These lectures present a relatively recent introduction into the class of discontinuos Galerkin (DG) methods, named Discontinuous Petrov-Galerkin (DPG) methods. DPG methods, in which DG spaces form a critical ingredient, can be thought of as least-square methods in nonstandard norms, or as Petrov-Galerkin methods with special test spaces, or as a nonstandard mixed method. We will pursue all the...

In this paper we consider the local discontinuous Galerkin method based on the generalized alternating numerical fluxes, for solving the linear convection-diffusion equations in one dimension and two dimensions. As an application of generalized Gauss–Radau projections, we get rid of the dual argument and obtain directly the optimal L2-norm error estimate in a uniform framework. The sharpness of...

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R, approximated by a discontinuous Galerkin method with polynomials of any degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. ...

We consider linear first order scalar equations of the form ρt + div(ρv) + aρ = f with appropriate initial and boundary conditions. It is shown that approximate solutions computed using the discontinuous Galerkin method will converge in L[0, T ;L(Ω)] when the coefficients v and a and data f satisfy the minimal assumptions required to establish existence and uniqueness of solutions. In particula...

Article history: Received 22 July 2016 Received in revised form 18 September 2016 Accepted 3 October 2016 Available online 24 October 2016