Convergence analysis of linear or quadratic X-FEM for curved free boundaries

Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis

“…Numerical tests are carried out in Section 6, and numerical orders are compared with their theoretical counterparts. Based on these tests, some theoretical predictions appear suboptimal, down h 1/2 , but the reason for this appears documented by . We show, for a quadratic displacement, that the combination of a quadratic geometrical description of the interface with the new restriction algorithm yields optimal convergence in practice , with an order 2.…”

“…The approach of Legay and coworkers with curved quadratic triangles as subcells was chosen here. In an earlier paper , the accuracy of this description was quantified and extensively discussed. In this section, we recall the procedure and some important results about its accuracy.…”

“…Appendix A of may be consulted for a proof of this point and a definition of geometric continuity. Hence, φ is the signed distance to the interface and ∇ φ , which verifies |∇ φ |=1, and gives the normal direction to the interface at the projection point (see Figure ).…”

“…Let us now give some background evaluation from about the accuracy of both approximation stages (see Table and Figure ). As for the first stage, it holds under regularity assumptions on Γ (, Lemma 4.1) …”

On the construction of approximation space to model discontinuities and cracks with linear and quadratic extended finite elements

“…The case of linear F (x) is slightly different, due to its link with the geometrical approximation of the level sets and more particularly of the interface established at the end of "Enrichment functions for weak discontinuities" section. Ferté et al [45,46] showed for strong discontinuities that the approximation of the level set impacts the expected theoretical convergence rates, in the case of quadratic elements: 2 is expected if a quadratic approximation of the level set is used (equivalently quadratic F (x)), while 1.5 is expected if a linear representation of the level set is used (equivalently linear F (x)). To illustrate this behavior, we also considered two models based on the enrichment designed for strong discontinuities, using Lagrange multipliers to enforce the continuity of the displacement through the interface: one with quadratic shape functions and a linear approximation of the level set and one with quadratic shape functions and a quadratic approximation of the level set.…”