scholarly journals Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
L. Jones Tarcius Doss ◽  
A. P. Nandini
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Piecewise Linear ◽  
Quadrature Rule ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Splitting Technique ◽  
Element Method

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.

scholarly journals Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

2021 ◽  
Author(s):  
şuayip toprakseven ◽  
Peng Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Reaction ◽  
Stability And Convergence ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Weak Galerkin ◽  
Order Error ◽  
Galerkin Finite Element ◽  
Element Method

In this paper, a modified weak Galerkin finite element method on Shishkin mesh has been developed and analyzed for the singularly perturbed convection-diffusion-reaction problems. The proposed method is based on the idea of replacing the standard gradient (derivative) and convection derivative by modified weak gradient (derivative) and modified weak convection derivative, respectively, over piecewise polynomials of degree $k\geq1$. The present method is parameter-free and has less degree of freedom compared to the weak Galerkin finite element method. Stability and convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy norm are proved. The method is uniformly convergent, i.e., the results hold uniformly regardless of the value of the perturbation parameter. Numerical experiments confirm these theoretical findings on Shishkin meshes. The numerical examples are also carried out on B-S meshes to confirm the theoretical results. Moreover, the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2-$ norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L_\infty-$ norm.

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