Error estimates for the finite-element solution of an elliptic singularly perturbed problem

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

“…This condition has been used in many papers and books on uniform convergence analysis, such as [2,17,18,21]. The existence and uniqueness of the solutions to both the continuous and the finite element problem do not need this condition, but it will be used in the proof of the error estimates in Section 5.…”

“…We also require that the mesh points on Γ match those from T (1) h . This mesh is denoted by T (2) h . Without loss of generality, we assume that the vertices in T (2) h not on ∂Ω are numbered from N 1 + 1 to N 1 + N 2 .…”

“…Without loss of generality, we assume that the vertices in T (2) h not on ∂Ω are numbered from N 1 + 1 to N 1 + N 2 . The set of edges of T (2) h not on ∂Ω is denoted by E (2) h . Obviously all rectangles in Ω (1) 2 and Ω…”

“…The first dual mesh, denoted by D (2) h , is the Dirichlet tessellation associated with the mesh nodes in T (2) h , i.e., the element d i ∈ D (2) h associated with the node x i of T (2) h is given by…”

“…To overcome this stability problem, many methods have been proposed. These include upwind methods (cf., for example, [5,11,12,4,2]), streamline diffusion methods (cf., for example, [13]) and exponentially fitted methods (cf., for example, [14,15,18,23,24]). However, no method guarantees, in general, that a numerical solution converges to the exact one uniformly in ε on an unstructured triangular partition.…”