Biharmonic Problem Research Articles

Solving the singular two-dimensional fourth order problem by the mortar spectral element method

In this work, we implement the mortar spectral element method for the biharmonic problem with a homogeneous boundary condition. We consider a polygonal domain with corners which relies on the mortar decomposition domain technique. We propose the Strang and Fix algorithm, which permits to enlarge the discrete space of the solution by the first singular function. The interest of this algorithm is the approximation of the solution and the leading singular coefficient which has a physical significance in the propagation of cracks. We give some numerical results which confirm the optimality of the order of the error.

Adaptive Morley element algorithms for the biharmonic eigenvalue problem

This paper is devoted to the adaptive Morley element algorithms for a biharmonic eigenvalue problem in mathbb{R}^{n} (ngeq2). We combine the Morley element method with the shifted-inverse iteration including Rayleigh quotient iteration and the inverse iteration with fixed shift to propose multigrid discretization schemes in an adaptive fashion. We establish an inequality on Rayleigh quotient and use it to prove the efficiency of the adaptive algorithms. Numerical experiments show that these algorithms are efficient and can get the optimal convergence rate.

Isogeometric Schwarz preconditioners for the biharmonic problem

A scalable overlapping Schwarz preconditioner for the biharmonic Dirichlet problem discretized by isogeometric analysis is constructed, and its convergence rate is analyzed. The proposed preconditioner is based on solving local biharmonic problems on overlapping subdomains that form a partition of the CAD domain of the problem and on solving an additional coarse biharmonic problem associated with the subdomain coarse mesh. An h-analysis of the preconditioner shows an optimal convergence rate bound that is scalable in the number of subdomains and is cubic in the ratio between subdomain and overlap sizes. Numerical results in 2D and 3D confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the isogeometric polynomial degree p and regularity k.

Existence of solutions for a class of p(x)-biharmonic problems without (A-R) type conditions

Existence of solutions for a class of p(x)-biharmonic problems without (A-R) type conditions

Compact difference scheme with high accuracy for one dimensional unsteady quasi-linear biharmonic problem of second kind: Application to physical problems

Compact difference scheme with high accuracy for one dimensional unsteady quasi-linear biharmonic problem of second kind: Application to physical problems

Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. III---Discontinuous Galerkin and Other Interior Penalty Methods

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix--Raviart interior penalty method for linear elasticity, and the quadratic $C^0$ interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases.

Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems

The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto–Sivashinsky equation and extended Fisher–Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.

The fully nonconforming virtual element method for biharmonic problems

In this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

A mixed finite element method for a sixth-order elliptic problem

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits natural nested discretization. Then, we present multi-level finite element schemes by implementing the algorithm as in Lin and Xie (Math Comput 84:71–88, 2015) to the nested discretizations on a series of nested grids. The multi-level mixed scheme for the biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

Existence of solutions for a nonhomogeneous p(x)-biharmonic problem

This article is devoted to study the existence of solutions of nonlinear problem of fourth order governed by p(x)-biharmonic operator under certain conditions on f. Our technical approach is based on the Minty-Browder theorem applied to compact operators and operators of (S+) type.

Multiple solutions to a class of p(x)-biharmonic differential inclusion problem with no-flux boundary condition

In this paper, we obtain the existence of at least two nontrivial solutions for a p(x)-biharmonic differential inclusion problem with no-flux boundary condition. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

Applications of topolological methods to the semilinear biharmonic problem with different powers

We prove the existence of multiple solutions for the fourth order nonlinear elliptic problem with fully nonlinear term. Our method is based on the critical point theory; the variation of linking method and category theory.

Integral Equation Formulation of the Biharmonic Dirichlet Problem

We present a novel integral representation for the biharmonic Dirichlet problem. To obtain the representation, the Dirichlet problem is first converted into a related Stokes problem for which the Sherman-Lauricella integral representation can be used. Not all potentials for the Dirichlet problem correspond to a potential for Stokes flow, and vice-versa, but we show that the integral representation can be augmented and modified to handle either simply or multiply connected domains. The resulting integral representation has a kernel which behaves better on domains with high curvature than existing representations. Thus, this representation results in more robust computational methods for the solution of the Dirichlet problem of the biharmonic equation and we demonstrate this with several numerical examples.

C 0IPG adaptive algorithms for the biharmonic eigenvalue problem

This paper focuses on C 0IPG adaptive algorithms for the biharmonic eigenvalue problem with the clamped boundary condition. We prove the reliability and efficiency of the a posteriori error indicator of the approximating eigenfunctions and analyze the reliability of the a posteriori error indicator of the approximating eigenvalues. We present two adaptive algorithms, and numerical experiments indicate that both algorithms are efficient.

Representation formula and bi-Lipschitz continuity of solutions to inhomogeneous biharmonic Dirichlet problems in the unit disk

The aim of this paper is twofold. First, we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic Dirichlet problems, and then prove the bi-Lipschitz continuity of the solutions.

English

The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate.

Minimization of inhomogeneous biharmonic eigenvalue problems

Abstract Biharmonic eigenvalue problems arise in the study of the mechanical vibration of plates. In this paper, we study the minimization of the first eigenvalue of a simplified model with clamped boundary conditions and Navier boundary conditions with respect to the coefficient functions which are of bang-bang type (the coefficient functions take only two different constant values). A rearrangement algorithm is proposed to find the optimal coefficient function based on the variational formula of the first eigenvalue. On various domains, such as square, circular and annular domains, the region where the optimal coefficient function takes the larger value may have different topologies. An asymptotic analysis is provided when two different constant values are close to each other. In addition, a symmetry breaking behavior is also observed numerically on annular domains.

Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition

We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.

Biharmonic Equation on Annulus in a Unit Sphere with Polynomial Boundary Condition

We studied Biharmonic boundary value problem on annulus with polynomial data. We utilize an exact algorithms for solving Laplace equations with Dirichlet conditions with polynomial functions data. The algorithm requires differentiation of the boundary function, but no integration.