Polygonal Domains Research Articles

High-dimensional finite elements for multiscale Maxwell-type equations

We consider multiscale Maxwell-type equations in a domain |$D\subset\mathbb{R}^d$| (⁠|$d=2,3$|⁠), which depend on |$n$| microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in |$\mathbb{R}^{(n+1)d}$|⁠. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in |$\mathbb{R}^d$|⁠. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution |$u_0$| of the homogenized problem to belong to |$H^1({\rm curl\,},D)$|⁠. However, in polygonal domains, |$u_0$| belongs only to a weaker regularity space |$H^s({\rm curl\,},D)$| for |$0<s<1$|⁠. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to |$H^{1+s}(D)$| (standard procedure requires |$H^2(D)$| regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results.

Regularity estimates and optimal finite element methods in [formula omitted] on polygonal domains

Consider the Poisson equation with the Dirichlet boundary condition on a bounded convex polygonal domain Ω⊂R2. We investigate the finite element approximation of singular solutions that are due to the non-smoothness of the domain in the Wp1 norm (1<p≤∞). In particular, with analysis in weighted Sobolev spaces and weighted Hölder spaces, we provide regularity requirements on the given data and specific parameter-selection criteria for graded meshes, such that the resulting numerical approximation achieves the optimal convergence rate in Wp1. Sample results from various numerical tests are provided to confirm the theory.

Convergence of natural adaptive least squares finite element methods

The first-order div least squares finite element methods provide inherent a posteriori error estimator by the elementwise evaluation of the functional. In this paper we prove Q-linear convergence of the associated adaptive mesh-refining strategy for a sufficiently fine initial mesh with some sufficiently large bulk parameter for piecewise constant right-hand sides in a Poisson model problem. The proof relies on some modification of known supercloseness results to non-convex polygonal domains plus the flux representation formula. The analysis is carried out for the lowest-order case in two-dimensions for the simplicity of the presentation.

Spanners for Geodesic Graphs and Visibility Graphs

Let \(\mathcal{P}\) be a set of n points inside a polygonal domain \(\mathcal{D}\). A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set \(\mathcal{P}\) with respect to the geodesic distance function \(\pi \) where for any two points p and q, \(\pi (p,q)\) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., \(h=0\)), we construct a (\(\sqrt{10}+\epsilon \))-spanner that has \(O(n \log ^2 n)\) edges. For a case where there are h holes, our construction gives a (\(5+\epsilon \))-spanner with the size of \(O(n\sqrt{h}\log ^2 n)\). Moreover, we study t-spanners for the visibility graph of \(\mathcal{P}\) (\(VG(\mathcal{P})\), for short) with respect to a hole-free polygonal domain \(\mathcal{D}\). The graph \(VG(\mathcal{P})\) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (\(3+\epsilon \))-spanner of size \(O(n^{4/3+\delta })\) for some \(\delta >0\). In addition, we show that there is a set \(\mathcal{P}\) of n points such that any \((3-\epsilon )\)-spanner of \(VG(\mathcal{P})\) must contain \(\varOmega (n^2)\) edges.

Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

In this article, we study numerical approximation of eigenvalue problems of the Schrodinger operator $-\Delta u + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two-dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several subdomains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities a...

Formation of modulated phases and domain rigidification in fatty acid-containing lipid membranes.

We investigated the phase behavior of lipid membranes containing fatty acids (FAs) by microscopy and differential scanning calorimetry. We used palmitic acid (saturated FA), oleic acid (cis-isomer of unsaturated FA), elaidic acid (trans-isomer of unsaturated FA), and phytanic acid (branched FA) and examined the effects of FAs on phase-separated structures in lipid bilayer membranes consisting of dioleolylphosphocholine (DOPC)/dipalmitoylphosphocholine (DPPC)/cholesterol (Chol). Palmitic acid and elaidic acid exclude Chol from the DPPC-rich phase. As a result, the liquid-ordered phase formed by DPPC and Chol transforms into a solid-ordered phase. Oleic acid and phytanic acid significantly reduce the line tension at the liquid domain boundary. This decrease in line tension leads to the formation of modulated phases, such as striped, hexagonal, and polygonal domains. We measured the line tension and the interdomain interaction in these specific domains by an image analysis. The result showed that oleic acid and phytanic acid-containing vesicles as well as palmitic acid-containing vesicles are not spherical, and this domain-induced deformation is explained theoretically.

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in \cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.

GUE corners limit of $q$-distributed lozenge tilings

We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider each tiling is weighted proportionally to $q^{\mathsf{vol} }$, where $\mathsf{vol} $ is the volume under the corresponding 3D stepped surface. We prove the following Interlacing Central Limit Theorem: as $q\rightarrow 1$, the domain gets large, and the fixed top row approximates a given nonrandom profile, the vertical lozenges are distributed as the eigenvalues of a GUE random matrix and of its successive principal corners. Our results extend the GUE corners asymptotics for tilings of bounded polygonal domains previously known in the uniform (i.e., $q=1$) case. Even though $q$ goes to $1$, the presence of the $q$-weighting affects non-universal constants in our Central Limit Theorem.

Numerical analysis of heat conduction problems on irregular domains by means of a collocation meshless method

A Least Squares Collocation Meshless Method based on Radial Basis Function (RBF) interpolation is used to solve steady state heat conduction problems on 2D polygonal domains using MATLAB® environment. The point distribution process required by the numerical method can be fully automated, taking account of boundary conditions and geometry of the problem to get higher point distribution density where needed. Several convergence tests have been carried out comparing the numerical results to the corresponding analytical solutions to outline the properties of this numerical approach, considering various combinations of parameters. These tests showed favorable convergence properties in the simple cases considered: along with the geometry flexibility, these features confirm that this peculiar numerical approach can be an effective tool in the numerical simulation of heat conduction problems.

An All Quadrilateral Automesh Generation Technique and Explicit Integration Scheme for Finite Elements to Solve Some Elliptic Boundary Value Problems in Two Space

This paper proposes the explicit integration scheme for a unique linear convex quadrilateral which can be obtained from an arbitrary linear triangle by joining the centroid to the midpoints of sides of the triangle. The explicit integration scheme proposed for these unique linear convex quadrilaterals is derived by using the standard transformations in two steps. We first map an arbitrary triangle into a standard right isosceles triangle by using an affine linear transformation from global (x, y) space into a local space (u, v). We discretise the standard right isosceles triangle in (u, v) space into three unique linear convex quadrilaterals. We have shown that any unique linear convex quadrilateral in (x, y) space can be mapped into one of the unique quadrilaterals in (u, v) space. We can always map these linear convex quadrilaterals into a 2-aquare in a natural space by an appropriate bilinear transformation. Using these two mappings, we have established an integral derivative product relation between the linear convex quadrilaterals in the (x, y) space interior to the arbitrary triangle and the linear convex quadrilaterals of the local space (u, v) interior to the standard right isosceles triangle. Further , we have shown that the product of global derivative integrals in (x, y) space can be expressed as a matrix triple product P ( X (2 * area of the arbitrary triangle in (x, y) space ) in which P is a geometric properties matrix and is the product of global derivative integrals in (u, v) space.We have shown that the explicit integration of the product of local derivative integrals in (u, v) space over the unique quadrilateral spanning vertices (1/3 ,1/3), (0, ½), (0, 0) , (1/2, 0) is now possible by use of symbolic processing capabilities in MATLAB which are based on Maple – V software package. In a similar manner we have found explicit integrals of ihe shape function and global derivative products as well as the product of shape functions.We use these integral values in computing stiffness matrix for some elliptic equations with constant coefficients.The proposed explicit integration scheme is a useful technique for boundary value problems governed by either a single equation or a system of second order partial differential equations. The physical applications of such problems are numerous in science, and engineering, the examples of single equations are the well known Laplace and Poisson equations with suitable boundary conditions. These problems are already studied in authors recent paper. We have demonstrated the proposed explicit integration scheme to solve the elliptic equations with constant coefficients subject to Dirichlet boundary conditions over simple polygonal domains like 1-square and 2-square.We have demonstrated the propsed scheme and technique to study five typical elliptic boundary value problems.Three FEM models with 2400,5400 and 7776 elements and 2481,5521 and 7921 nodes respectively are used.Tables of fem computed values,theoretical values at selected nodes and the finite element meshes are appended.The findings confirm with surface plots of exact solutions for the considered elliptic boundary value problems. Key words: Explicit Integration, Finite Element Method, Matlab Symbolic Mathematics, All Quadrilateral Mesh Generation Technique,Elliptic equations with constant coefficients ,Dirichlet Boundary Conditions ,Polygonal Domain,2-square,1-square

Computing the $$L_1$$ Geodesic Diameter and Center of a Polygonal Domain

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the $$L_1$$L1 geodesic diameter in $$O(n^2+h^4)$$O(n2+h4) time and the $$L_1$$L1 geodesic center in $$O((n^4+n^2 h^4)\alpha (n))$$O((n4+n2h4)ź(n)) time, respectively, where $$\alpha (\cdot )$$ź(·) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in $$O(n^{7.73})$$O(n7.73) or $$O(n^7(h+\log n))$$O(n7(h+logn)) time, and compute the geodesic center in $$O(n^{11}\log n)$$O(n11logn) time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on $$L_1$$L1 shortest paths in polygonal domains.

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017

Mathematical and numerical analysis of the time-dependent Ginzburg–Landau equations in nonconvex polygons based on Hodge decomposition

We prove well-posedness of the time-dependent Ginzburg–Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in $H^1(\Omega )$ in general, and so the finite element method (FEM) may give incorrect solutions. To overcome this difficulty, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit $H^1$ solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on the proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.

Bulk Universality for Random Lozenge Tilings Near Straight Boundaries and for Tensor Products

We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformly random lozenge tilings of large polygonal domains on triangular lattice and to the probability measures describing the decomposition in Gelfand-Tsetlin bases of tensor products of representations of unitary groups. In a weaker form our theorem also applies to random domino tilings.

A unified solution approach for a large variety of antiplane shear and torsion notch problems: Theory and examples

In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to be tackled. The method is based on the complex potential approach for antiplane elasticity combined with the use of proper conformal mappings. In particular, it is shown that a well defined analytical link does exist between the complex potential to be used to determine stresses and the first derivative of the conformal mapping used to mathematically describe the notch profile. This makes some methodologies such as Schwarz-Christoffel transformation, which allows describing any polygonal domain automatically, very attractive for the direct solution of notch problems. A bulk of solutions are provided to support this finding, from cracks and pointed notches, to radiused notches. In addition, the accuracy of each proposed solution is discussed in detail taking advantage of a bulk of results from FE analyses.

Numerical integration of arbitrary functions over a convex and non-convex polygonal domain by quadrature method

The objective of this paper is to present some of our recent development in numerical integration of arbitrary function over a convex and non-convex polygonal domain are approximated by quadrature method based on Generalized Gaussian quadrature rule, to evaluate the typical integrals governed by the proposed method.

Some remarks on a priori estimates of highly regular solutions for the Poisson equation in polygonal domains

This paper presents two expressions for \(H^3\) and \(H^4\) semi-norms for the solutions of the Poisson equation in two-dimensional polygonal domains. These equalities enable us to obtain higher order constructive a priori error estimates for finite element approximation of the Poisson equation with validated computing.

Boundary layer analysis of nonlinear reaction–diffusion equations in a polygonal domain

We propose a boundary layer analysis which fits a domain with corners. In particular, we consider nonlinear reaction–diffusion problems posed in a polygonal domain having a small diffusive coefficient ε>0. We present the full analysis of the singular behaviours at any orders with respect to the parameter ε where we use a systematic nonlinear treatment initiated in Jung et al. (2016). The boundary layers are formed near the polygonal boundaries and two adjacent ones overlap at a corner P and the overlapping produces additional layers, the so-called corner layers. It is noteworthy that the boundary layers are also degenerate due to the singularities of the solutions involving a negative power of the radial distance to the corner P which are present in the Laplace operator on a sector (sector corresponding to the part of the polygon near the corner). The corner layers are then designed to absorb both the singularities and the interaction of the two boundary layers at P.

Finite Element Analysis Of Linear Elastic Torsion For Regular Polygons

This paper presents an explicit finite element integration scheme to compute the stiffness matrices for linear convex quadrilaterals. Finite element formulationals express stiffness matrices as double integrals of the products of global derivatives. These integrals can be shown to depend on triple products of the geometric properties matrix and the matrix of integrals containing the rational functions with polynomial numerators and linear denominator in bivariates as integrands over a 2-square. These integrals are computed explicitely by using symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains.We have also developed an automatic all quadrilateral mesh generation technique for convex polygonal domain which provides the nodal coordinates and element connectivity.We have demonstrated the proposed explicit integration scheme to solve the Poisson Boundary Value Problem for a linear elastic torsion of a non-circular bar with cross sections having profiles of equilateral triangle, a square and regular polygons (pentagon(5-gon)to icosagon(20gon)) which are inscribed in a circle of unit radius. Monotonic convergence from below is observed with known analytical solutions for the Prandtl stress function and torsional constant .We have shown the solutions in Tables which list both the FEM and exact solutions. The graphical solutions of contour level curves and the corresponding finite element meshes are also displayed.

Finite Element Solution of Poisson Equation over Polygonal Domains using an Explicit Integration Scheme and a novel Auto Mesh Generation Technique

Finite Element Solution of Poisson Equation over Polygonal Domains using an Explicit Integration Scheme and a novel Auto Mesh Generation Technique