Solution Of Interface Problems Research Articles

Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem II: A simplified algorithm

The solution of interface problems with non-smooth interfaces, having corners connected by straight lines, exhibit weak singularity along the interface and may have singularities at the corners. In an earlier paper Zhang et al. (2019), a Strongly Stable Generalized Finite Element Method (SSGFEM) was proposed to approximate solution of such problems. The method required two separate enrichment functions at O(h−1) number of nodes (h is the mesh parameter) and a local orthogonalization technique, requiring additional operations, was used to ensure its stability and robustness. In this paper, we present a simplified SSGFEM that does not require two enrichment functions at any of the nodes. As a result, we obtain an optimally converging GFEM that is stable, i.e., its scaled condition number is of the same order that of a standard FEM, and robust, i.e., the conditioning is independent of the position of meshes relative to the interface; these properties are illuminated by well-designed experiments. The simplified SSGFEM does not require LOT and requires less degrees of freedom, and is easier to implement, in contrast to the SSGFEM proposed in (Zhang et al., 2019). We also proved the optimal convergence for the simplified SSGFEM in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner.

ANALYSIS OF OPTICAL WAVEGUIDES WITH ARBITRARY INDEX PROFILE USING AN IMMERSED INTERFACE METHOD

A numerical technique is described that can efficiently compute solutions of interface problems. These are problems with data, such as the coefficients of differential equations, discontinuous or even singular across one or more interfaces. A prime example of these problems are optical waveguides, and as such the scheme is applied to Maxwell's equations as they are formulated to describe light confinement in Bragg fibers. It is based on standard finite differences appropriately modified to take into account all possible discontinuities across the waveguide's interfaces due to the change of the refractive index. Second- and fourth-order schemes are described with additional adaptations to handle matrix eigenvalue problems, demanding geometries and defects.

A mathematical model and numerical solution of interface problems for steady state heat conduction

We study interface (or transmission) problems arising in the steady state heat conduction for layered medium. These problems are related to the elliptic equation of the form Au : = −∇(k(x)∇u(x)) = F(x), x ∈ Ω ⊂ ℝ2, with discontinuous coefficient k = k(x). We analyse two types of jump (or contact) conditions across the interfaces and of the layered medium Ω : = Ω1 ∪ Ωδ ∪ Ω2. An asymptotic analysis of the interface problem is derived for the case when the thickness (2δ > 0) of the layer (isolation) Ωδ tends to zero. For each case, the local truncation errors of the used conservative finite difference scheme are estimated on the nonuniform grid. A fast direct solver has been applied for the interface problems with piecewise constant but discontinuous coefficient k = k(x). The presented numerical results illustrate high accuracy and show applicability of the given approach.

An application of quasi-Newton methods for the numerical solution of interface problems

Quasi-Newton methods are applied to solve interface problems which arise from domain decomposition methods. These interface problems are usually sparse systems of linear or nonlinear equations. We are interested in applying these methods to systems of linear equations where we are not able or willing to calculate the Jacobian matrices as well as to systems of nonlinear equations resulting from nonlinear elliptic problems in the context of domain decomposition. Suitability for parallel implementation of these algorithms on coarse-grained parallel computers is discussed.

Solution of Interface Problems by Homogenization. III

The paper is the third in a series. The first two, Solution of interface problems by homogenization I,II are devoted to the study of the linear case. This, paper studies the case of strongly nonlinear differential equations.

Solution of Interface Problems by Homogenization. II

Solution of Interface Problems by Homogenization. II

Solution of Interface Problems by Homogenization. I

The problem of an elliptic differential equation with nonsmooth coefficients is studied. It is assumed that the coefficients are double periodic with the period H and the right-hand side is smooth. The behavior of the solution for $H \to 0$ is investigated and theorems about limiting solution are proved. The study is related to the homogenization ideas used in an intuitive way in the theory of nuclear reactors and theory of composite materials.