Biharmonic Problem Research Articles

On the robust solution of an isogeometric discretization of bilaplacian equation by using multigrid methods

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H2-conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases.

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.

Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson’s problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.

Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method

Abstract We describe and analyze the weighted extended b-spline (WEB-Spline) mesh-free finite element method for solving the p-biharmonic problem. The WEB-Spline method uses weighted extended b-splines as basis functions on regular grids and does not require any mesh generation which eliminates a difficult, time consuming preprocessing step. Accurate approximations are possible with relatively low-dimensional subspaces. We perform some numerical experiments to demonstrate the efficiency of the WEB-Spline method.

A Cartesian Compact Scheme for the Navier–Stokes Equations in Streamfunction Formulation in Irregular Domains

In Ben-Artzi et al. (SIAM J Numer Anal 47:3087–3108, 2009) we introduced an embedded Cartesian scheme for the biharmonic problem in two dimensions. Here we extend this methodology to the 2D Navier–Stokes system. Hermite (or Birkhoff) interpolation is invoked in one and two dimensions to obtain finite difference operators. The consistency analysis of the discrete formulas for irregular grids is emphasized. Numerical results demonstrate remarkable accuracy for a series of test cases for flows in elliptical domains.

Spectral-Galerkin approximation and optimal error estimate for biharmonic eigenvalue problems in circular/spherical/elliptical domains

In this paper, we propose and analyze spectral-Galerkin methods for the biharmonic eigenvalue problem in circular/spherical/elliptical domains. We first analyze the eigenfunction formulated fourth-order equation under the polar coordinates, then we derive the pole condition and reduce the problem on a circular disk/sphere to a sequence of equivalent one-dimensional eigenvalue problems that can be solved in parallel. The novelty of our approach lies in the construction of suitably weighted Sobolev spaces according to the pole conditions, based on which, the optimal error estimate for approximated eigenvalue of each one-dimensional problem can be obtained. Further, we extend our method to the non-separable biharmonic eigenvalue problem in an elliptic domain and establish the optimal error bounds. Finally, we provide some numerical experiments to validate our theoretical results and algorithms.

A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems

In this paper, we express and analyze mixed discontinuous Galerkin(DG) methods of biharmonic eigenvalue problems as well as present the error analysis for them. The analysis consists of two parts. First, we propose a residual-based a posteriori error estimator in the approximate eigenfunctions and eigenvalues. The error in the eigenfunctions is measured both in the L2 and DG (energy-like) norms. In addition, we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so, the computed eigenvalue converges to a true eigenvalue. Next, we establish an a priori error estimate with the optimal convergence order both in the L2 and DG norms. We show that the methods can retain the same convergence properties they enjoy in the case of source problems.

A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.

Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More

We prove a comprehensive solution theory using tools from functional analysis, show corresponding variational formulations, and present functional a posteriori error estimates for general linear first order systems of typefor two densely defined and closed (possibly unbounded) linear operators A1 and A2 having the complex property . As a prototypical application we will discuss the system of electro-magneto statics in 3D with mixed tangential and normal boundary conditionsOur theory covers a lot more applications in 2D, 3D, and ND, such as general differential forms and all kind of systems arising, e.g., in general relativity, biharmonic problems, Stokes equations, or linear elasticity, to mention just a few, for exampleall with possibly mixed boundary conditions of generalized tangential and normal type. Second order systems of typeswill be considered as well using the same techniques.

A new two-level implicit scheme based on cubic spline approximations for the 1D time-dependent quasilinear biharmonic problems

In this article, we present a new two-level implicit cubic spline numerical method of accuracy 2 in time and 4 in spatial direction for the numerical solution of 1D time-dependent quasilinear biharmonic equation subject to appropriate initial and natural boundary conditions prescribed. The easiness of the proposed numerical method lies in their 3-point discretization in which we use two points $$ x \pm (h/2) $$ and a central point ‘x’ in spatial direction. Using the continuity of the first-order derivative of cubic spline function, we derive the fourth-order accurate numerical method for the time-dependent biharmonic equation on a uniform mesh. The stability consideration of the proposed method is discussed using a model linear problem. The proposed cubic spline method successfully implements on generalized Kuramoto–Sivashinsky and extended Fisher–Kolmogorov equations. From the numerical experiments, we obtain better computational results compared to the results discussed in earlier research work.

Solving boundary problems for biharmonic operator by using Integro-differential operators of fractional order

The investigation of the properties of the integro-differential operators will be carried out. Which generalizes the well-known Bavrin operators to the fractional value of the parameters. The properties of the defined operators are in the classes of the polyharmonic operators. It is established that the newly defined fractional operators map the polyharmonic functions on the ball to the polyharmonic functions. Also it is proposed that the inverse for the fractional operator and application of the integro-differential fractional operators to solve biharmonic problems with fractional boundary conditions. The sufficient condition for existence and uniqueness of the solution for biharmonic equation with fractional boundary conditions are obtained. The solution of the biharmonic equation is obtained by using the integro-differential fractional operator.

On the Mixed Dirichlet--Farwig biharmonic problem in exterior domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Dirichlet--Farwig biharmonic problem in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight $|x|^a$. Admitting different boundary conditions, we used the variation principle and depending on the value of the parameter $a$, we obtained uniqueness (non-uniqueness) theorems of the problem or present exact formulas for the dimension of the space of solutions.

A quadratic spline collocation method for the Dirichlet biharmonic problem

A new method based on quadratic spline collocation is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two second-order partial differential equations. This method involves the solution of an auxiliary biharmonic problem using fast Fourier transforms and the solution of a nonsymmetric Schur complement system using preconditioned BICGSTAB, at a total cost of $N^{2} \log N$ on an N × N uniform partition of the unit square. The results of numerical experiments demonstrate the optimality of the global accuracy of the method and also superconvergence results, in particular, third-order accuracy in the $L^{\infty }$ norm of the solution and its fourth-order accuracy at the partition nodes and the collocation points.

On the Mixed Dirichlet–Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

We studied the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we studied the unique solvability of the mixed Dirichlet–Steklov-type and Steklov-type biharmonic problems in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight | x | a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

The biharmonic homotopy problem for unit vector fields on 2-tori

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.

High accuracy two-level implicit compact difference scheme for 1D unsteady biharmonic problem of first kind: application to the generalized Kuramoto–Sivashinsky equation

ABSTRACTWe propose a two-level implicit high order compact scheme for the one-dimensional unsteady biharmonic problem of first kind. The values of φ and are prescribed on the boundary. Using a combination of values of φ and at each grid point, the difference formula is derived for the unsteady biharmonic equation without discretizing the boundary conditions. The proposed method has second order time accuracy and fourth order space accuracy using just three grid points of a single compact stencil at every time level. The first order space derivative is also computed with same accuracy as a by-product of the method. Using the Von-Neumann analysis, the derived scheme is shown to be unconditionally stable. With a slight modification, the proposed method is applicable to solve singular problems. The performance of the proposed scheme is illustrated by numerical experiments done on a collection of test problems having physical significance including the nonlinear Kuramoto–Sivashinsky equation.

Formula omitted]-FEM for a stabilized three-field formulation of the biharmonic problem

In this paper, a three-field formulation for the biharmonic equation is considered that consists of the potential u, the flux σ and a Lagrange multiplier. The use of stabilization techniques circumvents the discrete inf–sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields. A priori error estimates as well as reliable and efficient a posteriori error estimates are derived. The proof of the latter is based on an implicit, never to be computed, H2-reconstruction of the discrete potential. Several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p-robust with an efficiency index tending towards the optimal value of one. The numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the Lagrange multiplier for the a priori error estimate is not fulfilled. In particular, h- and hp-adaptivity recover optimal algebraic and exponential converge rates, respectively.

Two-level implicit high order method based on half-step discretization for 1D unsteady biharmonic problems of first kind

In this work, compact difference scheme based on half-step discretization is proposed to solve fourth order time dependent partial differential equations subject to Dirichlet and Neumann boundary conditions. The difference method reported here is second order accurate in time and fourth order accurate in space. The scheme employs only three grid points at each time-level and the given boundary conditions are exactly satisfied with no further approximations at the boundaries. The linear stability of the presented method has been examined and it is shown that the proposed two-level finite difference method is unconditionally stable for a model linear problem. The developed method is directly applicable to fourth order parabolic partial differential equations with singular coefficients which is the main highlight of our work. The method is successfully tested on singular problem. The proposed method is applied to find the numerical solution of the classical nonlinear Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation. Comparison of the obtained results with those of some earlier known methods demonstrate the superiority of the present approach.

On fast solution of the primal biharmonic problem

On fast solution of the primal biharmonic problem

Quasi-Optimal Nonconforming Methods for Symmetric Elliptic Problems. II---Overconsistency and Classical Nonconforming Elements

We devise variants of classical nonconforming methods for symmetric elliptic problems. These variants differ from the original ones only by transforming discrete test functions into conforming functions before applying the load functional. We derive and discuss conditions on these transformations implying that the ensuing method is quasi-optimal and that its quasi-optimality constant coincides with its stability constant. As applications, we consider the approximation of the Poisson problem with Crouzeix--Raviart elements and higher order counterparts and the approximation of the biharmonic problem with Morley elements. In each case, we construct a computationally feasible transformation and obtain a quasi-optimal method with respect to the piecewise energy norm on a shape regular mesh.