Search

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]

The multidimensional positive definite advection transport algorithm (MPDATA) is an important numerical method for the computation of atmospheric dynamics. MPDATA is second-order accurate, positive definite, conservative, and computationally efficient. However, the method is problematic in which it results in a loss of precision when computing a nonuniform irregular grid. Furthermore, research revealed two reasons for this problem. On the one hand, numerical discretization of boundary derivatives of the finite-volume method is incompatible with nonuniform meshes (or grids); on the other hand, the up-wind scheme of staggered grids is not applicable to the calculation of irregular grids. We overcome these two problems by using the multipoint Taylor expansion method to obtain a boundary derivative numerical approximation scheme that does not depend on the grid structure. Furthermore, combined with the well-balance central-upwind scheme, a positive definite advection scheme for irregular meshes is proposed. Then, the positivity of the new numerical scheme is analyzed. Finally, the result of this study is verified by numerical simulation. [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 this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three-dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable [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: 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]

In this research paper, a weighed DG finite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general finite element meshes. The efficiency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and flexibility of the method. [ABSTRACT FROM AUTHOR]

The objective of this work is to study finite element methods for approximating the solution of convection equations on surfaces embedded in R 3 . We propose the discontinuous Galerkin (DG) isogeometric analysis (IgA) formulation to solve convection problems on implicitly defined surfaces. Three numerical experiments shows that the numerical scheme converges with the optimal convergence order. [ABSTRACT FROM AUTHOR]