Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales: proofs

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters $0< \epsilon \le \mu \le 1$, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have \emph{boundary layers} which overlap and interact, based on the relative size of $\epsilon$ and $$. We construct full asymptotic expansions together with error bounds that cover the complete range $0 < \epsilon \leq \mu \leq 1$. For the present case of analytic input data, we derive derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order

Simulation of failure of air plasma sprayed thermal barrier coating due to interfacial and bulk cracks using surface-based cohesive interaction and extended finite element method

This article describes a method of predicting the failure of a thermal barrier coating system due to interfacial cracks and cracks within bulk coatings. The interfacial crack is modelled by applying cohesive interfaces where the thermally grown oxide is bonded to the ceramic thermal barrier coating. Initiation and propagation of arbitrary cracks within coatings are modelled using the extended finite element method. Two sets of parametric studies were carried out, concentrating on the effect of thickness of the oxide layer and that of initial cracks within the ceramic coating on the growth of coating cracks and the subsequent failures. These studies have shown that a thicker oxide layer creates higher tensile residual stresses during cooling from high temperature, leading to longer coating cracks. Initial cracks parallel to the oxide interface accelerate coating spallation, and simulation of this process is presented in this article. By contrast, segmented cracks prevent growth of parallel cracks which can lead to spallation

A high frequency hphp boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods

Scalable numerical approach for the steady-state ab initio laser theory

We present an efficient and flexible method for solving the non-linear lasing equations of the steady-state ab initio laser theory. Our strategy is to solve the underlying system of partial differential equations directly, without the need of setting up a parametrized basis of constant flux states. We validate this approach in one-dimensional as well as in cylindrical systems, and demonstrate its scalability to full-vector three-dimensional calculations in photonic-crystal slabs. Our method paves the way for efficient and accurate simulations of lasing structures which were previously inaccessible.Comment: 17 pages, 8 figure

Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

The upper triangular solutions to the three-state constant quantum Yang-Baxter equation

In this article we present all nonsingular upper triangular solutions to the constant quantum Yang-Baxter equation $R_{j_1j_2}^{k_1k_2}R_{k_1j_3}^{l_1k_3}R_{k_2k_3}^{l_2l_3}= R_{j_2j_3}^{k_2k_3}R_{j_1k_3}^{k_1l_3}R_{k_1k_2}^{l_1l_2}$ in the three state case, i.e. all indices ranging from 1 to 3. The upper triangular ansatz implies 729 equations for 45 variables. Fortunately many of the equations turned out to be simple allowing us to start breaking the problem into smaller ones. In the end we had a total of 552 solutions, but many of them were either inherited from two-state solutions or subcases of others. The final list contains 35 nontrivial solutions, most of them new.Comment: 24 Pages in LaTe

General DG-Methods for Highly Indefinite Helmholtz Problems

We develop a stability and convergence theory for a Discontinuous Galerkin formulation (DG) of a highly indefinite Helmholtz problem in $$\mathbb R ^{d}$$ R d , $$d\in \{1,2,3\}$$ d ∈ { 1 , 2 , 3 } . The theory covers conforming as well as non-conforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the DG-method admits a unique solution under much weaker conditions. As an application we present the error analysis for the $$hp$$ hp -version of the finite element method explicitly in terms of the mesh width $$h$$ h , polynomial degree $$p$$ p and wavenumber $$k$$ k . It is shown that the optimal convergence order estimate is obtained under the conditions that $$kh/\sqrt{p}$$ kh / p is sufficiently small and the polynomial degree $$p$$ p is at least $$O(\log k)$$ O ( log k ) . On regular meshes, the first condition is improved to the requirement that $$kh/p$$ kh / p be sufficiently smal

Permutation-type solutions to the Yang-Baxter and other n-simplex equations

We study permutation type solutions to n-simplex equations, that is, solutions whose R matrix can be written as a product of delta- functions depending linearly on the indices. With this ansatz the D^{n(n+1)} equations of the n-simplex equation reduce to an [n(n+1)/2+1]x[n(n+1)/2+1] matrix equation over Z_D. We have completely analyzed the 2-, 3- and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.Comment: 20 pages, LaTeX2e. to appear in J. Phys. A: Math. Gen. (1997