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In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

Two‐dimensional biharmonic boundary‐value problems are considered by the linear barycentric rational method, the unknown function was approximated by the barycentric rational function. For the biharmonic equation, we change the biharmonic equation into the two Poisson equations by depression of order. The linear equations of discrete the biharmonic equation was changed into matrix form. For the basis of barycentric rational function, we present the convergence rate of linear barycentric rational method for biharmonic equation by depression of order. At last, several numerical examples are provided to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

One of the reasons for the success of the finite element method in Solid Mechanics, among other Applied Sciences, is its versatility to deal with bodies of arbitrary shape. In case the problem at hand is modeled by second‐order partial differential equations with Dirichlet conditions prescribed on a curvilinear boundary, method's isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for polygonal or polyhedral domains and methods of order greater than one based on standard straight‐edged elements. However, besides algebraic and geometric inconveniences, the isoparametric technique is limited in scope, since its extension to degrees of freedom other than function values is not straightforward. In previous work the author introduced a simple alternative, which bypasses the above drawbacks without eroding qualitative approximation properties. Among other advantages, it can do without curved elements and is based only on polynomial Algebra. The purpose of this paper is two‐fold: On the one hand the mathematical foundations of this method are established, in the framework of a model Poisson problem solved with Lagrange elements; in particular optimal‐rate error estimates both in energy norm and in the mean‐square sense are demonstrated and a numerical comparison with the isoparametric technique is carried out. On the other hand this study is extended to Hermite elements with normal‐derivative degrees of freedom for solving the equation modeling the bending of clamped thin plates with a curvilinear edge. [ABSTRACT FROM AUTHOR]

The laminar viscous fluid flow past a circular cylinder placed in a square cavity of a uniform cross-section is generated by applying injection/suction at the adjacent sidewalls. The temperature on the side walls without suction and on the boundary of the cylinder is kept constant, and constant heat flux is maintained on the walls with suction. The streamline flow pattern and isothermal lines are drawn. The flow is assumed to be Stokesian. Hence, the resulting biharmonic equation is solved for stream function by expressing it in two coupled equations, and a 5-point formula is used to solve these equations. Fictitious nodes are introduced for derivative boundary conditions for stream function by using a central difference scheme, and a 3-point backward difference formula is used for derivative boundary conditions on temperature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill‐posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well‐posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in ℒ2 uniformly with respect to the space coordinate under some a priori assumptions on the solution. [ABSTRACT FROM AUTHOR]

The purpose for this paper is to study heat transfer in a rectangular channel with suction applied at the adjacent two side walls. A two‐dimensional laminar viscous fluid flow is generated due to the application of suction/injection. The other two opposite sides are kept at a constant temperatures, and the walls with suction are maintained at a constant heat flux. The streamlines thus obtained due to the flow and isothermal lines and heat function are analyzed. The regions of high and low frictions are found by drawing contours of the entropy generation number and the Bejan number. The biharmonic equation for stream function is numerically solved by the Finite Difference Method (FDM) using a 13‐point formula, and a five‐point formula is used to solve for all other harmonic equations for temperature, heat function, and pressure. For derivative boundary conditions, the central difference formula with fictitious nodes is used. In the analysis, we note that the corner points are regions of high energy dissipation. The least dissipation of energy is near the wall, where the nondimensional temperature is 1. This paper analyzes heat transfer in the rectangular channel through the heat function and entropy generation number. [ABSTRACT FROM AUTHOR]

Let Hom+(T) be the class of all sense‐preserving homeomorphic self‐mappings of T={z=x+iy∈C:|z|=1}. The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω)  =  g in unit disk D with associated boundary value conditions Δω|T=φ∈C(T) and ω|T=f∗∈Hom+(T). Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K′,||φ||∞:=supz∈D|φ(z)| and ||g||∞:=supz∈D|g(z)| are small enough. [ABSTRACT FROM AUTHOR]

We consider the problem Δ2u = V(x)up + ϵ in RN with u,Δu→0 as |x|→ + ∞, where p=N+4N−4, N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation: BvΔ2u+Vλ(x)u=α(x)f(u)+νK(x)|u|q−2uinRN,u∈H2(RN),where N⩾5,ν ∈ (0,ν0],1  <  q  <  2,Δ2u  =  Δ(Δu) and Vλ(x)  =  λa(x) − b(x) with λ > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem Bν. [ABSTRACT FROM AUTHOR]

Highlights are the following: For any integer n ≥ 0 , we construct C n ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions. To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly. To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions. Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains. [ABSTRACT FROM AUTHOR]

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown. An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation [ABSTRACT FROM AUTHOR]

We consider a commutative algebra [ABSTRACT FROM AUTHOR]

In this article, we present a method to obtain a C1-surface, defined on a bounded polygonal domain Ω, which interpolates a specific dataset and minimizes a certain 'energy functional.' The minimization space chosen is the one associated to the Powell-Sabin finite element, whose elements are C1-quadratic splines. We develop a general theoretical framework for that, and we consider two main applications of the theory. For both of them, we give convergence results, and we present some numerical and graphical examples. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 798-821, 2015 [ABSTRACT FROM AUTHOR]

The algorithm for the solution of the first biharmonic problem by the Trefftz method is presented. The solution is purely mathematical and follows Mikhlin [29] approach to the solution of the harmonic problem. As such it fills in the gap in the approximate methods of Mathematical Physics. Validity of the algorithm is demonstrated on two examples. The boundary values of the sought function and its normal derivative must be given explicitly to achieve correct results by the algorithm. For that reason, the presented algorithm is, however, not adequate for the plane problems of the theory of elasticity or for plates with free or simply supported edges. [ABSTRACT FROM AUTHOR]

The paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond_eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al. (Eng. Anal. Boundary Elements 2004; 28:79–96; Trefftz and Collocation Methods. WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond_eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond_eff is more advantageous than Cond. Besides since the bounds of Cond_eff and Cond involve the estimation of the minimal singular value σmin of the discrete matrix F, and since the estimation of σmin is challenging and difficult, the proof for lower bounds of σmin in this paper is important and intriguing. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner-Fox-Schmit rectangle, the generalization of the Bramble-Pasciak-Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived. [ABSTRACT FROM AUTHOR]