A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.

翻译：针对二维空间中的奇异摄动椭圆问题，构造了一种Petrov-Galerkin有限元方法。该解包含一个规则边界层和两个特征边界层。在一个坐标方向上采用指数样条作为测试函数，并与定义在Shishkin网格上的双线性试验函数相结合。结果表明，该数值方法是一种稳定的参数一致数值方法，与在同一种网格上采用迎风格式相比，实现了更高的收敛阶。