A new parameter free partially penalized immersed finite element and the optimal convergence analysis

Abstract

This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously even though the curved interface is approximated by line segments. The new method and analysis have also been extended to problems with variable coefficients. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.Keywords interface problem • partially penalized immersed finite element • unfitted mesh • trace inequality • optimal error estimates of IFEM

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

“…Now we prove that the result is also valid for arbitrary triangles in the following lemma. Note that for the IFEs using nodal values as degrees of freedom, the maximum angle condition, α max ≤ π/2 on interface triangles, is necessary to ensure the unisolvence (see [16]). This property of the unisolvence of basis functions is one of advantages of nonconforming IFEs compared with the IFEs using nodal values as degrees of freedom.…”

“…First, we prove that Crouzeix-Raviart IFE basis functions are unisolvent on arbitrary triangles. Our recent study in [16] shows that, for the IFEs using nodal values as degrees of freedom, the maximum angle condition, α max ≤ π/2 on interface triangles, is necessary to ensure the unisolvence of the basis functions. The unisolvence of basis functions on arbitrary triangles is a significant advantage of the nonconforming IFEs using integral-value degrees of freedom over the IFEs using nodal values as degrees of freedom.…”

An immersed $CR$-$P_0$ element for Stokes interface problems and the optimal convergence analysis

“…Note that for many existing IFEs developed for other interface problems, the unisolvence of IFE shape functions with respect to the degrees of freedom relies on the mesh assumption, i.e., the noobtuse angle condition [14,25,16,23]. Recently, we showed that for second-order elliptic interface problems, if integral-values on edges are used as the degrees of freedom, then the unisolvence holds on arbitrary triangles [24].…”

“…The other type of unfitted mesh methods is the immersed finite element (IFE) method [30,33], which modifies the traditional finite element on interface elements according to interface conditions to achieve the optimal approximation capability, while keeping the degrees of freedom unchanged. For second-order elliptic interface problems, IFE methods have been studied extensively in [32,20,34,15,25]. However, for the Stokes interface problems, there are much fewer works on IFE method in the literature.…”

An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes

“…One drawback of the IFE space is that it does not belong to H(div; Ω) because the normal components of IFE functions may be discontinuous across the edges cut by the interface (called interface edges). One approach to overcome the nonconformity is to add consistent terms locally on interface edges to the bilinear form, and therefore a penalty term should also be included simultaneously to ensure the stability (see [30,25]). However, for the lowest-order Raviart-Thomas mixed finite element method, we find that the consistent term is zero.…”