This paper deals with convergence theorems of the Galerkin finite element approximation for second-order elliptic boundary value problems. Under some quite general settings, we show not only pointwise but also prove that norm approximate operator converges to corresponding inverse a linear operator. Since estimates linearized play an essential role in numerical verification method solutions non...

In this article, using the weighted discrete least-squares, we propose a patch reconstruction finite element space with only one degree of freedom per element. As approximation space, it is applied to discontinuous Galerkin methods upwind scheme for steady-state convection-diffusion-reaction problems over polytopic meshes. The optimal error estimates are provided in both diffusion-dominated and...

We consider the numerical approximation of the spectrum of a secondorder elliptic eigenvalue problem by the hybridizable discontinuous Galerkin (HDG) method. We show for problems with smooth eigenfunctions that the approximate eigenvalues and eigenfunctions converge at the rate 2k + 1 and k + 1, respectively. Here k is the degree of the polynomials used to approximate the solution, its flux, an...

We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L2 with the optimal order of k + 1,...

In the framework of abstract linear inverse problems in infinite-dimensional Hilbert space we discuss generic convergence behaviours approximate solutions determined by means general projection methods, namely outside standard assumptions Petrov–Galerkin truncation schemes. This includes a discussion mechanisms why error or residual generically fail to vanish norm, and identification practicall...

We investigate an ultraweak variational formulation for (parameterized) linear differential-algebraic equations with respect to the time variable which yields optimally stable system. This is used within a Petrov-Galerkin method derive certified detailed discretization provides approximate solution in setting as well model reduction spirit of Reduced Basis Method. A computable sharp error bound...

and Applied Analysis 3 nonlinear term is treated with the Chebyshev collocation method. The time discretization is a classical Crank-Nicholson-leap-frog scheme. Yuan and Wu 43 extended the Legendre dual-Petrov-Galerkin method proposed by Shen 44 , further developed by Yuan et al. 45 to general fifth-order KdV-type equations with various nonlinear terms. The main aim of this paper is to propose ...

We consider a general system of n1 semilinear parabolic partial differential equations and n2 ordinary differential equations, with locally Lipschitz continuous nonlinearities. We analyse the well-posedness of this problem, exploiting the tools of the semigroups theory, and derive other further regularity results and conditions for the boundedness of the solution. We define the Galerkin semidis...

The subject of the paper is the derivation and analysis of third order nite volume evolution Galerkin schemes for the two-dimensional wave equation system. To achieve this the rst order approximate evolution operator is considered. A recovery stage is carried out at each level to generate a piecewise polynomial approximation ~ U n = RhU n 2 S h from the piecewise constantU 2 S h , to feed into ...

In this paper, we propose a new hybridized discontinuous Galerkin method for the convection-diffusion-reaction problems with mixed boundary conditions. The coercivity of the convection-reaction part is achieved by adding an upwinding term. We give error estimates of optimal order in the piecewise H1-seminorm. Furthermore, we show that the approximate solution of our scheme is close to that of t...