Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA First-order Two-scale Analysis for Contact Problems with Small Periodic Configurations
186
0
0.0
(
0
)
Added by
Changqing Ye
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
This paper is devoted to studying a type of contact problems modeled by hemivariational inequalities with small periodic coefficients appearing in PDEs, and the PDEs we considered are linear, second order and uniformly elliptic. Under the assumptions, it is proved that the original problem can be homogenized, and the solution weakly converges. We derive an $O(epsilon^{1/2})$ estimation which is pivotal in building the computational framework. We also show that Robin problems--- a special case of contact problems, it leads to an $O(epsilon)$ estimation in $L^2$ norm. Our computational framework is based on finite element methods, and the numerical analysis is given, together with experiments to convince the estimation.
rate research
Read More
This paper presents a new fast multipole boundary element method (FM-BEM) for solving the acoustic transmission problems in 2D periodic media. We divide the periodic media into many fundamental blocks, and then construct the boundary integral equations in the fundamental block. The fast multipole algorithm is proposed for the square and hexagon periodic systems, the convergence of the algorithm is analyzed. We then apply the proposed method to the acoustic transmission problems for liquid phononic crystals and derive the acoustic band gaps of the phononic crystals. By comparing the results with those from plane wave expansion method, we conclude that our method is efficient and accurate.
Using the framework of operator or Cald{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.
This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrodinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension $d$, under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.
The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a modified energy norm, as well as a reduced convergence order of ${cal O}(h^{3/2})$ in the standard $H^1$-norm. Finally, we present numerical examples to substantiate the theoretical findings.
A general analysis framework is presented in this paper for many different types of finite element methods (including various discontinuous Galerkin methods). For second order elliptic equation, this framework employs $4$ different discretization variables, $u_h, bm{p}_h, check u_h$ and $check p_h$, where $u_h$ and $bm{p}_h$ are for approximation of $u$ and $bm{p}=-alpha abla u$ inside each element, and $ check u_h$ and $check p_h$ are for approximation of residual of $u$ and $bm{p} cdot bm{n}$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
