Summary: ROBUSTNESS OF A SPLINE ELEMENT METHOD WITH
Abstract. The spline element method with constraints is a discretization method
where the unknowns are expanded as polynomials on each element and Lagrange
multipliers are used to enforce the interelement conditions, the boundary conditions
and the constraints in numerical solution of partial differential equations. Spaces of
piecewise polynomials with global smoothness conditions are known as multivariate
splines and have been extensively studied using the Bernstein-Bezier representation
of polynomials. It is used here to write the constraints mentioned above as linear
equations. In this paper, we illustrate the robustness of this approach on two
singular perturbation problems, a fourth order problem and a Stokes-Darcy flow.
It is shown that the method converges uniformly in the perturbation parameter.
In this paper, we use piecewise polynomials of arbitrary degree to approximate the
solutions of two singular perturbation problems. It is shown that the method, called
here spline element method, and described in , , , ,  and  is robust
with respect to the perturbation parameter. We consider plane problems but the
method extends easily to 3D problems.