Arbitrary Plane Domain Research Articles

Application of a modified manifold-based exponentially convergent algorithm to solve elliptic boundary-value problems

In this paper, we combine a novel finite difference method (FDM) with a manifold-based exponentially convergent algorithm (MBECA) to solve a nonlinear elliptic boundary-value problem defined in an arbitrary plane domain. It is very difficult to solve nonlinear and geometric complexity problems by conventional FDM. To overcome these problems, the concepts of internal residual and boundary residual in a fictitious rectangular domain are introduced. Besides, by adding a fictitious time coordinate, we avoid the need to treat complex boundary conditions but only use shape functions. Also, it is not necessary to solve an inverse matrix of algebraic equations when by using the MBECA. Moreover, in order to increase the numerical stability of the MBECA, we introduce a group-preserving scheme (GPS) to address fictitious time integration. Given the cone structure of the GPS and MBECA and their manifold properties, we can preserve the manifold path on the cone structure by a weighting factor such that the MBECA must also exhibit a cone construction, Lie algebra and group properties at each fictitious time step. Finally, the accuracy and convergence behaviour of this present method are demonstrated in several examples.

Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

In this paper, the inverse Cauchy problem for Laplace equation defined in an arbitrary plane domain is investigated by using the collocation Trefftz method (CTM) with a better postconditioner. We first introduce a multiple-scale Rk in the T-complete functions as a set of bases to expand the trial solution. Then, the better values of Rk are sought by using the concept of an equilibrated matrix, such that the resulting coefficient matrix of a linear system to solve the expansion coefficients is best-conditioned from a view of postconditioner. As a result, the multiple-scale Rk can be determined exactly in a closed-form in terms of the collocated points used in the collocation to satisfy the boundary conditions. We test the present method for both the direct Dirichlet problem and the inverse Cauchy problem. A significant reduction of the condition number and the effective condition number can be achieved when the present CTM is used, which has a good efficiency and stability against the disturbance from large random noise, and the computational cost is much saving. Some serious cases of the inverse Cauchy problems further reveal that the unknown data can be recovered very well, although the overspecified data are provided only at a 20% of the overall boundary.

Isoperimetric inequality for torsional rigidity in multidimensional domains

We consider the Saint-Venant functional P for the torsional rigidity in an arbitrary plane or spatial domain. The main result of this paper is the sharp estimate P ≤ (4/n)m, where n is the dimension of the space, and m is the harmonic mean of moments of inertia of the domain about the coordinate planes. The extremal domains are ellipsoids of a special kind. Thus, we obtain a generalization of the isoperimetric inequality proved by E. Nicolay for the torsional rigidity of simply connected plane domains.

A Highly Accurate MCTM for Inverse Cauchy Problems of Laplace Equation in Arbitrary Plane Domains

We consider the inverse Cauchy problems for Laplace equation in simply and doubly connected plane domains by recoverning the unknown bound- ary value on an inaccessible part of a noncircular contour from overspecified data. A modified Trefftz method is used directly to solve those problems with a simple collocation technique to determine unknown coefficients, which is named a mod- ified collocation Trefftz method (MCTM). Because the condition number is small for the MCTM, we can apply it to numerically solve the inverse Cauchy problems without needing of an extra regularization, as that used in the solutions of direct problems for Laplace equation. So, the computational cost of MCTM is very sav- ing. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown boundary data, even by subjecting the given data to a large noise. Keyword: Inverse Cauchy problem, Modified Trefftz method, Laplace equation, Modified collocation Trefftz method (MCTM)

A Highly Accurate MCTM for Direct and Inverse Problems of Biharmonic Equation in Arbitrary Plane Domains

Trefftz method (TM) is one of widely used meshless numerical methods in el- liptic type boundary value problems, of which the approximate solution is expressed as a lin- ear combination of T-complete bases, and the un- known coefficients are determined from bound- ary conditions by solving a linear equations sys- tem. However, the accuracy of TM is severely limitedbyitsill-conditioning. Thispaperisacon- tinuation of the work of Liu (2007a). The col- location TM is modified and applied to the di- rect and inverse problems of biharmonic equation in a simply connected plane domain. Due to its well-conditioningoftheresultinglinearequations system, the present modified collocation Trefftz method (MCTM) can effectively solve the inverse problems without needing of overspecified data, iteration, and regularization. So that, the compu- tational cost of MCTM is saving. Numerical ex- amples show the effectiveness of the new method in providing highly accurate numerical solutions even subjectingto large noise of the given bound- ary data.

Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation

The method of fundamental solu- tions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are deter- mined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its ill-conditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Tre- fftz method and MFS for circular domain, while the second proposed a modified Trefftz method (MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the well-posedness of MTM, we can alleviate the ill-conditioning of MFS through a new linear equations system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Nu- merical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10 −10 . The present method is fully not similar to the preconditioning technique as used to solve the ill-conditioned lin- ear equations system.

A Highly Accurate Solver for the Mixed-Boundary Potential Problem and Singular Problem in Arbitrary Plane Domain

A highly accurate new solver is de- veloped to deal with interior and exterior mixed- boundary value problems for two-dimensional Laplace equation, including the singular ones. To promote the present study, we introduce a circu- lar artificial boundary which is uniquely deter- mined by the physical problem domain, and de- rive a Dirichlet to Robin mapping on that arti- ficial circle, which is an exact boundary condi- tion described by the first kind Fredholm integral equation. As a consequence, we obtain a modi- fied Trefftz method equipped with a characteristic length factor, ensuring that the new solver is sta- ble because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the Fourier coefficients. We find that the new method is powerful even for the problem with complex boundary shape and with adding random noise on the boundary data. It is also applicable to the sin- gular problem of Motz type, resulting to a highly accurate result never seen before. Keyword: Laplace equation, Artificial circle, DtR mapping, Modified Trefftz method, Mixed- boundary value problem, Singular problem

Lower estimates for the number of closed trajectories of generalized billiards

The mathematical study of periodic billiard trajectories is a classical question that goes back to George Birkhoff. A billiard is the motion of a particle in the absence of field of force. Trajectories of such a particle are geodesics. A billiard ball rebounds from the boundary of a given domain making the angle of incidence equal the angle of reflection. Let k be a fixed integer. Birkhoff proved a lower estimate for the number of closed billiard trajectories of length k in an arbitrary plane domain. We give a general definition of a closed billiard trajectory when the billiard ball rebounds from a submanifold of a Euclidean space. Besides, we show how in this case one can apply the Morse inequalities using the natural symmetry (a closed polygon may be considered starting at any of its vertices and with the reversed direction). Finally, we prove the following estimate. Let M be a smooth closed m-dimensional submanifold of a Euclidean space, and let p > 2 be a prime integer. Then M has at least $$\frac{{(B - 1)((B - 1)^{p - 1} - 1)}}{{2p}} + \frac{{mB}}{2}(p - 1)$$ closed billiard trajectories of length p. Bibliography: 7 titles.

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρi(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε‐domain perturbation considered in this paper.

A Proof of the Uniformization Theorem for Arbitrary Plane Domains

We present a simple constructive proof of the Uniformization Theorem which works for plane domains. The proof is a combination of covering space theory and Koebe’s constructive proof of the Riemann mapping theorem, and the resulting algorithm can be used to estimate the Poincaré metric for the domain.

A proof of the uniformization theorem for arbitrary plane domains

We present a simple constructive proof of the Uniformization Theorem which works for plane domains. The proof is a combination of covering space theory and Koebe’s constructive proof of the Riemann mapping theorem, and the resulting algorithm can be used to estimate the Poincaré metric for the domain.

A scheme for the automatic generation of triangular finite elements

AbstractThis paper describes a method for the automatic triangulation of arbitrary multilateral plane domains. In addition, the method can be used in connection with that suggested by Zienkiewicz and Phillips4 for the subdivision of curved‐boundary domains. The method can be described as general fully automatic and computer oriented. A Fortran computer program has been prepared by the author. Output can be of interactive, graphical or alphanumerical form. The program has been applied to a number of cases. The resulting triangulations are always satisfactory, even when vigorous changes of mesh sizes are encountered.

On moduli of plane domains

It is well known that an arbitrary plane domain of finite connectivity can be mapped conformally onto an annulus minus a certain number of circular slits. The parameters defining such a canonical domain are studied in the context of Teichmüller theory. Let Ω \Omega be a plane domain bounded by m ⩾ 3 m \geqslant 3 continua. Denote by T ( Ω ) T(\Omega ) the reduced Teichmüller space of Ω \Omega and by R ( Ω ) R(\Omega ) the space of conformal equivalence classes of domains bounded, as Ω \Omega is, by m continua. A real analytic map from T ( Ω ) T(\Omega ) onto an open subset S ( Ω ) S(\Omega ) of a 3 m − 6 3m - 6 dimensional product of circles and lines is constructed. It is shown that the map T ( Ω ) → S ( Ω ) T(\Omega ) \to S(\Omega ) is a regular covering map. Finally, it is observed that there is a finite sheeted covering map S ( Ω ) → R ( Ω ) S(\Omega ) \to R(\Omega ) .

Asymptotic solution of eigenvalue problems

A method is presented for the construction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations. It is an adaptation to bounded domains of the method previously devised to deduce the corrected Bohr-Sommerfeld quantum conditions. When applied to the reduced wave equation in various domains for which the exact solutions are known, it yields precisely the asymptotic forms of those solutions. In addition it has been applied to an arbitrary convex plane domain for which the exact solutions are not known. Two types of solutions have been found, called the “whispering gallery” and “bouncing ball” modes. Applications have also been made to the Schrödinger equation.