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In this research article, a discontinuous Galerkin method with a weighted parameter θ and a penalty parameter γ is proposed for solving the first order hyperbolic equation. The key aim of this method is to design an error estimation for both a priori and a posteriori error analysis on general finite element meshes. It is also exposed to the reliability and effectiveness of both parameters in the order of convergence of the solutions. For a posteriori error estimation, residual adaptive mesh- refining algorithm is employed. A series of numerical experiments are illustrated that demonstrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we express and analyze mixed discontinuous Galerkin(DG) methods of biharmonic eigenvalue problems as well as present the error analysis for them. The analysis consists of two parts. First, we propose a residual-based a posteriori error estimator in the approximate eigenfunctions and eigenvalues. The error in the eigenfunctions is measured both in the L2 and DG (energy-like) norms. In addition, we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so, the computed eigenvalue converges to a true eigenvalue. Next, we establish an a priori error estimate with the optimal convergence order both in the L2 and DG norms. We show that the methods can retain the same convergence properties they enjoy in the case of source problems. [ABSTRACT FROM AUTHOR]

In this paper, employing ideas developed for conservation law equations such as the Lax–Friedrich-type and Godunov-type numerical fluxes, we describe the numerical schemes for approximating the solution of the limit problem arising in the homogenization of Hamilton–Jacobi equations. All approximation methods involve three steps. The first scheme is a provably monotonic discretization of the cell problem for approximating the effective Hamiltonian for a given vector P ∈ R N . Next, using interpolation, we present an approximation of the effective Hamiltonian in the domain R N . Finally, the numerical schemes of the Hamilton–Jacobi equations with the effective Hamiltonian approximation are constructed. We also present global error estimates including all the discrete mesh sizes. The theoretical results are illustrated through numerical examples, including two convex Hamiltonians and two non-convex Hamiltonians. [ABSTRACT FROM AUTHOR]

In the current paper, we derive the a priori error analysis ( hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature. [ABSTRACT FROM AUTHOR]

Abstract: In the current paper, we study the convergence properties of the DGFE approximation of optimal control problem governed by convection–diffusion equations. We derive a posteriori error estimates and a priori error estimates for both the states, ad-joint and the control variable approximation. For the optimal control problem, these estimates are apparently not available in the literature. [Copyright &y& Elsevier]

Abstract: We describe a technique for a posteriori error estimates suitable to the optimal control problem governed by the evolution equations solved by the method of lines. It is applied to the control problem governed by the parabolic equation, convection-diffusion equation and hyperbolic equation. The error is measured with the aid of the -norm in the space–time cylinder combined with a special time weighted energy norm. [ABSTRACT FROM AUTHOR]