Robin Boundary Value Problem Research Articles

Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension approach

Abstract A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\varOmega $ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $\varOmega _{h}$ and to address issues owing to the domain perturbation $\varOmega \neq \varOmega _{h}$. In contrast to the lift approach used in existing studies, we employ the extension approach, which need not assume that boundary nodes of $\partial \varOmega _{h}$ lie exactly on $\partial \varOmega $. Assuming that approximate domains and function spaces are given by isoparametric finite elements of order $k$, we prove the optimal rate of convergence in the $H^{1}$- and $L^{2}$-norms. A numerical example is given for the piecewise linear case $k = 1$.

A compact combination of second-kind Chebyshev polynomials for Robin boundary value problems and Bratu-type equations

AbstractThis paper presents a new algorithm for resolving linear and non-linear second-order Robin boundary value problems (BVPS) and the Bratu-type equations in one and two dimensions using spectral approaches. Basis functions according to second-kind shifted and modified shifted Chebyshev polynomials that comply with the Robin conditions are created. It has produced operational matrices for its derivatives. The provided solutions are the result of applying the collocation and tau approaches. These methods convert the problem dictated by its boundary conditions into a system of linear or non-linear algebraic equations that may be solved using any suitable numerical solver. Convergence analysis has been provided and it accords with the numerical results. Six numerical problems are provided to investigate and demonstrate the practical utility of the suggested method. The current results show that our method outperforms the previous methods in terms of accuracy which are presented in tables and figures.

3D Koch-type crystals

We consider the construction of a family \{K_N\} of 3 -dimensional Koch-type surfaces, with a corresponding family of 3 -dimensional Koch-type “snowflake analogues” \{\mathcal{C}_N\} , where N>1 are integers with N \not\equiv 0 \mathrm{mod}\,\,3 . We first establish that the Koch surfaces K_N are s_N -sets with respect to the s_N -dimensional Hausdorff measure, for s_N=\log(N^2+2)/\log(N) the Hausdorff dimension of each Koch-type surface K_N . Using self-similarity, one deduces that the same result holds for each Koch-type crystal \mathcal{C}_N . We then develop lower and upper approximation monotonic sequences converging to the s_N -dimensional Hausdorff measure on each Koch-type surface K_N , and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set \mathcal{C}_N . As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals \mathcal{C}_N , for N>2 .

Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions

A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, i.e. expectations with respect to the invariant law of RSDEs, both inside a domain in $\mathbb{R}^{d}$ and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.

Nitsche's method for a Robin boundary value problem in a smooth domain

AbstractWe prove several optimal‐order error estimates for a finite‐element method applied to an inhomogeneous Robin boundary value problem (BVP) for the Poisson equation defined in a smooth bounded domain in , . The boundary condition is weakly imposed using Nitsche's method. The Robin BVP is interpreted as the classical penalty method with the penalty parameter . The optimal choice of the mesh size relative to is a non‐trivial issue. This paper carefully examines the dependence of on error estimates. Our error estimates require no unessential regularity assumptions on the solution. Numerical examples are also reported to confirm our results.

Robin boundary-value problem for the Beltrami equation

UDC 517.5 We investigate the unique solution of the Robin boundary-value problem for the Beltrami equation with constant coefficients in the unit disc by using a technique based on a singular integral operator defined on L p ( 𝔻 ) for all p > 2.

Sufficient Condition for the Existence of an Additional Zone in Singularly Perturbated Second-Order Boundary Problem

Studies the Dirichlet, Neman and Robin boundary value problems for a singularly perturbed linear inhomogeneous second order ordinary differential equation. The considered boundary value problems have three features: the singular presence of a small parameter; the solution of the corresponding unperturbed equation has a k order pole and an additional boundary layer. The singular presence of a small parameter generates the classical boundary layer, and the singular point of the corresponding unperturbed equation generates the second boundary layer. As a result, we get a double boundary layer. A sufficient condition for the existence of an additional boundary layer is found.

Numerical algorithm with high accuracy for the modified Helmholtz equation with Robin boundary value problem

In order to obtain the numerical solutions of the modified Helmholtz equation with Robin boundary conditions in two dimensions, we have developed a numerical algorithm to solve the problem, which is based on a Nyström discretization. First, the problem is transformed to a weakly-singular integral equation. Then we construct a quadrature method that achieve a convergence rate of O(h3), which has the characteristics of simple calculation and high precision. Further, the convergence of the numerical solutions is proved based on the collectively compact operators theory and the single parameter asymptotic expansion of errors with odd power O(h3) is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. Finally, some numerical examples are presented to demonstrate the efficiency of the method.

Solving Robin condition using a method based on integral equation

In this paper, Robin boundary value problem is solved numerically with an integral equations’ based method. The suggested method has been compared with other existing methods or exact solution to approve the accuracy of this method.

A class of singularly perturbed Robin boundary value problems in critical case

This paper discusses a class of nonlinear singular perturbation problems with Robin boundary values in critical cases. By using the boundary layer function method and successive approximation theory, the corresponding asymptotic expansions of small parameters are constructed, and the existence of uniformly efficient smooth solutions is proved. Meanwhile, we give a concrete example to prove the validity of our results.

Sound radiation by a vibrating circular plate located at the bottom of a non-rigid flanged circular cylindrical tube

The sound transmission through a non-rigid circular tube has been investigated in this study. For the same, the Neumann–Robin boundary value problem has been solved. The problem deals with sound radiation by considering a complex system consisting of a fluid filled in the circular cylindrical tube having a finite impedance and the half-space. The outlet of the tube is embedded into a rigid infinite flat screen. The fluid inside the tube is disturbed by a vibrating circular plate embedded at the bottom of the tube. A coupled system of partial differential equations governing the vibration of the plate and the fluid has been solved. For this purpose, the superposition of the two solutions has been applied: one is the Neumann boundary value problem in the form of the Dini series and the other is the correction for the impedance boundary in the form of a Fourier series. The continuity conditions at the tube outlet have been expressed in terms of the modal coupling acoustic impedance coefficients. The use of the radial polynomials together with Cerjan’s expansion greatly improve the numerical analysis. Application of the additional Fourier series helps to avoid finding the complex eigenvalues. Truncation conditions in the Dini series and in the Fourier series have been applied to assure valid and accurate numerical results for frequencies up to 8kHz, a tube radius from 30 to 70 mm, and a tube length from 3 to 7 times the tube radius with an arbitrary excitation of the plate. The selected tube materials range from being acoustically soft, such as mineral wool, to acoustically hard, such as a porous material. The other advantage of using the Dini and the Fourier series is that they require only real eigenvalues, and thus it is not necessary to find the cumbersome complex eigenvalues of the tube. One more advantage of the rigorous solution is that it automatically satisfies the radiation condition in the half-space, whereas the finite elements method always satisfies it approximately. The findings of this study can be summarized as follows: The main computational cost is associated with a relatively significant number of the Fourier expansion terms. As a result, for frequencies smaller than 1.2kHz, the first result can be obtained rapidly using the finite elements method. In addition, it has been shown that using the radial polynomials to calculate the coupling integrals enables faster calculations by at least 120 times.

Bifurcation diagram of a Robin boundary value problem arising in MEMS

We consider a parabolic problem with Robin boundary condition which arises when the edge of a micro-electro-mechanical-system (MEMS) device is connected with a flexible nonideal support. Then via a rigorous analysis we investigate the structure of the solution set of the corresponding steady-state problem. We show that a critical value (the pull-in voltage) exists so that the system has exactly two stationary solutions when the applied voltage is lower than this critical value, one stationary solution for applying this critical voltage, and no stationary solution above the critical voltage.

Classification of Finite Morse Index Solutions for Robin Boundary Value Problems

Classification of Finite Morse Index Solutions for Robin Boundary Value Problems

Accumulation times for diffusion-mediated surface reactions

In this paper we consider a multiparticle version of a recent probabilistic framework for studying diffusion-mediated surface reactions. The basic idea of the probabilistic approach is to consider the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a totally reflecting boundary; the effects of surface reactions are then incorporated via an appropriate stopping condition for the local time. The propagator is determined by solving a Robin boundary value problem, in which the constant rate of reactivity is identified as the Laplace variable z conjugate to the local time, and then inverting the solution with respect to z. Here we reinterpret the propagator as a particle concentration in which surface absorption is counterbalanced by particle source terms. We investigate conditions under which there exists a non-trivial steady state solution, and analyze the relaxation to steady state by calculating the corresponding accumulation time. In particular, we show that the first two moments of the stopping local time density have to be finite.

Fitted mesh numerical method for a coupled system of singularly perturbed reaction-diffusion robin boundary value problem having boundary and internal layers

Fitted mesh numerical method for a coupled system of singularly perturbed reaction-diffusion robin boundary value problem having boundary and internal layers

A Robin boundary value problem for the Cauchy–Riemann operator in a ring domain

Abstract In this paper, we study a Robin condition for the inhomogeneous Cauchy–Riemann equation w z ¯ = f {w_{\bar{z}}=f} in a ring domain R, by reformulating it as a Dirichlet boundary condition.

Weighted global regularity estimates for elliptic problems with Robin boundary conditions in Lipschitz domains

Let n ≥ 2 and Ω be a bounded Lipschitz domain of R n . In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second-order elliptic equations of divergence form with real-valued, bounded, measurable coefficients on Ω. More precisely, let p ∈ ( n / ( n − 1 ) , ∞ ) . Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W 1 , p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1 , q estimates of solutions with q ∈ ( n / ( n − 1 ) , p ] and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second-order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, C 1 domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when n = 3 in case of bounded C 1 domains, also gives an alternative correct proof of some known result under an additional assumption. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces.

Numerical solution of third-order Robin boundary value problems using diagonally multistep block method

This numerical study exclusively focused on developing a diagonally multistep block method of order five (2DDM5) to get the approximate solution of the third-order Robin boundary value problems directly. The mathematical derivation of the developed 2DDM5 method is by approximating the integrand function with Lagrange interpolation polynomial. The proposed direct integrator scheme was applied to compute numerical solution at two-point concurrently. Shooting technique adapted with the Newtons divided difference interpolation method was implemented throughout the proposed algorithm. The theoretical characteristics of the developed method including the order, consistency, zero-stable and convergence are discussed. The method are tested on four Robin boundary value problems. The numerical results signify that the computational performances of the proposed method is competitive in terms of accuracy and efficiency than the existing method.

The conormal and Robin boundary value problems in nonsmooth domains satisfying a measure condition

We consider elliptic equations and systems in divergence form with the conormal or the Robin boundary conditions, with small BMO (bounded mean oscillation) or variably partially small BMO coefficients. We propose a new class of domains which are locally close to a half space (or convex domains) with respect to the Lebesgue measure in the system (or scalar, respectively) case, and obtain the Wp1 estimate for the conormal problem with the homogeneous boundary condition. Such condition is weaker than the Reifenberg flatness condition, for which the closeness is measured in terms of the Hausdorff distance, and the semi-convexity condition. For the conormal problem with inhomogeneous boundary conditions, we also assume that the domain is Lipschitz. By using these results, we obtain the Wp1 and weighted Wp1 estimates for the Robin problem in these domains.

Existence of solution in the bending of thin plates with Gurtin–Murdoch surface elasticity

We consider the well-posedness of classical boundary value problems in a theory of bending of thin plates which incorporates the effects of surface elasticity via the Gurtin–Murdoch surface model. We employ the fundamental solution of the governing system of equations to develop integral-type solutions of the corresponding Dirichlet, Neumann, and Robin boundary value problems. Using the boundary integral equation method, we subsequently establish results for the existence of a solution in the appropriate function spaces.