A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Finite element methods for semilinear elliptic problems with smooth interfaces

“…Remark 3.1 : The present work improves the results of [5] and [14] for elliptic interface problems under minimum regularity assumptions of the true solution and for a practical finite element discretization. To the best of authors knowledge Theorem 3.1 and Theorem 3.2 have not been established before for conforming finite element method with straight interface triangles.…”

“…Regarding the approximation of the bilinear form A h , we have the following result. For a proof, we refer to [14].…”

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

“…We now recall some existing results on the approximation A h and the inner product which will be frequently used in our analysis. For a proof, we refer to [16,11]. Lemma 2.1.…”

“…More recently, the authors of [10] have shown that the finite element solution converges to the exact solution at an optimal rate in L 2 and H 1 norms if the grid lines coincide with the actual interface by allowing interface triangles to be curved triangles. As it may be computationally inconvenient to fit the mesh to an arbitrary interface exactly, a finite element discretization based on [9] is considered in [11] for non-symmetric problems. Sub-optimal order error estimates in L 2 norm and optimal order error estimates in energy norm are shown in [11].…”

“…As it may be computationally inconvenient to fit the mesh to an arbitrary interface exactly, a finite element discretization based on [9] is considered in [11] for non-symmetric problems. Sub-optimal order error estimates in L 2 norm and optimal order error estimates in energy norm are shown in [11]. For detailed elaboration on finite element analysis for elliptic and parabolic interface problems, we refer to [12].…”