Uniform convergent modified weak Galerkin method for
convection-dominated two-point boundary value problems
Abstract
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.