Biharmonic Dirichlet Problem Research Articles

An Optimal Multilevel Method with One Smoothing Step for the Morley Element

Abstract In this paper, a class of multilevel preconditioning schemes is presented for solving the linear algebraic systems resulting from the application of Morley nonconforming element approximations to the biharmonic Dirichlet problem. Based on an appropriate space splitting of the finite element spaces associated with the refinements and the abstract Schwarz framework, we prove that the proposed multilevel methods with one smoothing step are optimal, i.e., the convergence rate is independent of the mesh sizes and mesh levels. Moreover, the computational complexity is also optimal since the smoothers are performed only once on each level in the algorithm. Numerical experiments are provided to confirm the optimality of the suggested methods.

On the explicit representation of the trace space H^{frac{3}{2}} and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

We consider the problem of describing the traces of functions in H^2(Omega ) on the boundary of a Lipschitz domain Omega of mathbb R^N, Nge 2. We provide a definition of those spaces, in particular of H^{frac{3}{2}}(partial Omega ), by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space H^1(Omega ), based on the classical second order Steklov problem.

An integral equation method for the Cahn-Hilliard equation in the wetting problem

We present an integral equation approach to solving the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. The discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, we split the solution into a volume potential computed with free space kernels, plus the solution to a second kind integral equation (SKIE). The volume potential is evaluated with the help of a box-based volume-FMM method. For non-box domains, the source density is extended by solving a biharmonic Dirichlet problem. The near-singular boundary integrals are computed using quadrature by expansion (QBX) with FMM acceleration. Our method has linear complexity in the number of surface/volume degrees of freedom and can achieve high order convergence in space with adaptive refinement to manage error from function extension.

A biharmonic converse to Krein–Rutman: a maximum principle near a positive eigenfunction

The Green function \(G_0(x,y)\) for the biharmonic Dirichlet problem on a smooth domain \(\Omega \), that is \(\Delta ^{2}u=f\) in \(\Omega \) with \( u=u_{n}=0 \) on \(\partial \Omega \), can be written as the difference of a positive function, which bears the singularity at \(x=y\), and a rank-one positive function, both of which satisfy the boundary conditions. See Grunau et al. (Proc Am Math Soc 139:2151–2161, 2011). More precisely \(G_0(x,y)= H(x,y)- c\, d(x)^2 d(y)^2\) holds, where \(d(\cdot )\) is the distance to the boundary \(\partial \Omega \) and where H contains the singularity and is positive. We will extend the corresponding estimates to \( G_{\lambda }(x,y)\) for the differential operator \(\Delta ^{2}-\lambda \) with an optimal dependence on \(\lambda \). As a consequence, strict positivity of an eigenfunction with a simple eigenvalue \(\lambda _{i}\) implies a positivity preserving property for \(\left( \Delta ^{2}-\lambda \right) u=f\) in \(\Omega \) with \(u=u_{n}=0\) on \(\partial \Omega \) for \(\lambda \) in a left neighbourhood of \(\lambda _{i} \). This result can be viewed as a converse to the Krein–Rutman theorem.

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.

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.

Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic Problem

We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretized by $C^1$ bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a $2\times 2$ block diagonal (BD) preconditioner and a block bordered diagonal (BBD) preconditioner. We prove mesh-independent bounds for the spectra of the BD-preconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite (SPD) and assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near-optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion, and nonconvex domains.

Solving the biharmonic Dirichlet problem on domains with corners

The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz' representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. Ciarlet and Raviart in and Monk in consider approaches through second order problems assuming that the domain is smooth. We will discuss what happens when the domain has corners. Moreover, we will suggest a setting, which is in some sense between Ciarlet‐Raviart and Monk, that inherits the benefits of both settings and that will give the weak solution through a system type approach.

Existence of Nonradial Solutions for Hénon Type Biharmonic Equation Involving Critical Sobolev Exponents

We prove the existence of nonradial solutions under some conditions for a semilinear biharmonic Dirichlet problem involving critical Sobolev exponents.

Asymptotic behavior of solutions of a biharmonic Dirichlet problem with large exponents

We analyze blow-up phenomena of bounded integrable solutions of a semilinear fourth order elliptic problem with a large exponent under Dirichlet boundary conditions. We extend the results obtained by Ren-Wei in [26] and [27] to the biharmonic case.

Matrix decomposition algorithms for elliptic boundary value problems: a survey

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 logN) on an N?×?N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two---point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson's equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems.

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W 2 2 . We show that the problem can be viewed as a generalization of the Dirichlet problem.

Three-dimensional image reconstruction using the PF/MFS technique

We propose a geometric modeling method in R 3 based on the so-called potential field (PF) modeling technique. The method is a new technique for surface reconstruction from a data set of scattered points taken on a surface. In this method, the construction of a geometric model is based on the solution of an elliptic boundary value problem determined using the method of fundamental solutions (MFS). We consider two such problems, Laplace's equation subject to Dirichlet boundary conditions and the biharmonic Dirichlet problem. We present the results of numerical experiments which demonstrate the efficacy of these approaches. We also identify differences and potential difficulties in the application of these approaches and propose ways of addressing them.

Subspace Interpolation with Applications to Elliptic Regularity

In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem and for the Stokes and the Navier–Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classic regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.

Orthogonal spline collocation methods for the stream function‐vorticity formulation of the Navier–Stokes equations

AbstractAn orthogonal spline collocation (OSC) spatial discretization is proposed for the solution of the fully coupled stream function‐vorticity formulation of the Navier–Stokes equations in two dimensions. For the time‐stepping, a three‐level leapfrog scheme is employed. This method is algebraically linear, and, at each time step, gives rise to a system of linear equations of the form arising in the OSC approximation of the biharmonic Dirichlet problem and can be solved by a fast direct method. Error estimates in the Hl–norm in space, l = 1,2, are derived for the semi‐discrete method and the fully‐discrete leapfrog scheme which is also shown to be second order accurate in time. Numerical results are presented which confirm the theoretical analysis and exhibit superconvergence phenomena, which provide superconvergent approximations to the components of the velocity. © John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N2 log2 N + mN2) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log2N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.

The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity

For a semilinear biharmonic Dirichlet problem in the ball with supercritical power-type nonlinearity, we study existence/nonexistence, regularity and stability of radial positive minimal solutions. Moreover, qualitative properties, and in particular the precise asymptotic behaviour near x = 0 for (possibly existing) singular radial solutions, are deduced. Dynamical systems arguments and a suitable Lyapunov (energy) function are employed.

A domain-type boundary-integral-equation method for two-dimensional biharmonic Dirichlet problem

This paper reports a new boundary-integral-equation method (BIEM) for numerically solving biharmonic problems with Dirichlet boundary conditions. For the solution of these problems in convex polygons, it was found that the accuracy of the conventional BIEM is significantly reduced, and spurious oscillatory behaviour is often observed in the boundary solutions especially for areas near corners (Mai-Duy N, Tanner RI. An effective high order interpolation scheme in BIEM for biharmonic boundary value problems. Eng Anal Bound Elem 2005;29:210–23). In this study, a new treatment for these difficulties is proposed. The unknown functions in boundary integrals are approximated using a domain-type interpolation scheme rather than traditional boundary-type interpolation schemes. Two test problems are considered to validate the formulation and to demonstrate the attractiveness of the proposed method.

Regularity in a Singular Biharmonic Dirichlet Problem

We consider the singular biharmonic equation with Dirichlet boundary conditions u = f0 and ?nu = f1 on . In our setup the boundary values fj (j = 0,1) are elements in two homogeneous Banach spaces Bj (j = 0,1) on . We give a sufficient condition on the spaces Bj (j = 0,1) to ensure that the solution u of this Dirichlet problem has the appropriate boundary values fj (j = 0,1) in the sense of convergence in spaces Bj (j = 0,1). Our results also apply in the unweighted case.

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the Bogner–Fox–Schmit rectangular element and the product two-point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006