Weak Galerkin finite element methods for fourth order singular perturbation problems

Abstract:

In this paper, we discuss the construction and analysis of the weak Galerkin (WG) finite element methods for the fourth order singular perturbation problems in two and three dimensions. By introducing the weak second order partial derivative operators, the WG method is constructed by adopting continuous piecewise polynomials of degree k>2 for the approximation to the displacement in the interior of elements, and discontinuous piecewise polynomials of degree k-1 for the approximations to the trace of displacement gradient on the inter-element boundaries. Based on the properties of the Scott-Zhang and L^2 projections, optimal error estimates in energy norm are derived. In addition, for the boundary layer case, we show that the methods are convergent uniformly with respect to the perturbation parameter. Numerical examples are provided to verify the theoretical results.