Higher-Order SGFEM for One-Dimensional Interface Elliptic Problems with Discontinuous Solutions

We study a class of enriched unfitted finite element or generalized finite element methods (GFEM) to solve a larger class of interface problems, that is, 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. The major challenge of GFEM development is to construct enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. The linear stable generalized finite element method (SGFEM) was recently developed using one enrichment function. We generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. So is the optimal order convergence at all nodes. To prove the efficiency of the SGFEM, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present

SoftIGA: soft isogeometric analysis

We extend the softFEM idea to isogeometric analysis (IGA) to reduce the stiffness (consequently, the condition numbers) of the IGA discretized problem. We refer to the resulting approximation technique as softIGA. We obtain the resulting discretization by first removing the IGA spectral outliers to reduce the system's stiffness. We then add high-order derivative-jump penalization terms (with negative penalty parameters) to the standard IGA bilinear forms. The penalty parameter seeks to minimize spectral/dispersion errors while maintaining the coercivity of the bilinear form. We establish dispersion errors for both outlier-free IGA (OF-IGA) and softIGA elements. We also derive analytical eigenpairs for the resulting matrix eigenvalue problems and show that the stiffness and condition numbers of the IGA systems significantly improve (reduce). We prove a superconvergent result of order $h^{2p+4}$ for eigenvalues where $h$ characterizes the mesh size and $p$ specifies the order of the B-spline basis functions. To illustrate the main idea and derive the analytical results, we focus on uniform meshes in 1D and tensor-product meshes in multiple dimensions. For the eigenfunctions, softIGA delivers the same optimal convergence rates as the standard IGA approximation. Various numerical examples demonstrate the advantages of softIGA over IGA