Showing papers on "Biharmonic equation published in 2014"

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

Abstract: A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete H2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L2 norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Abstract: We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L^∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L^2 norm of the source terms; its (pre-)computation involves minimizing O(H^(-d)) quadratic (cell) problems on (super-)localized sub-domains of size O(H ln(1/H)). The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for d ≤ 3, and polyharmonic for d ≥ 4, for the operator -div(a∇.) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method (O(H)) in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincare inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.

Guaranteed lower eigenvalue bounds for the biharmonic equation

Abstract: The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate's vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.

An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes

Abstract: This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H 2 -semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H 2 -equivalent norm for the WG finite element solutions. Error estimates in the usual L 2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

Biharmonic maps into a Riemannian manifold of non-positive curvature

Abstract: We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.

A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

Abstract: A \(C^0\)-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for \(C^0\) functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete \(H^2\) norm and the standard \(H^1\) and \(L^2\) norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order \((k+1-d)\) and the surface mass of order \((k+2-d)\) for the \(P_{k+2}\) finite element functions in \(d\)-dimensional space.

Application of the Generalized Finite-Difference Method to Inverse Biharmonic Boundary-Value Problems

Abstract: In this article, the generalized finite-difference method (GFDM), one kind of domain-type meshless method, is adopted for analyzing inverse biharmonic boundary-value problems. In inverse problems governed by fourth-order partial differential equations, overspecified boundary conditions are imposed at part of the boundary, and, on the other hand, part of the boundary segment lacks enough boundary conditions. The ill-conditioning problems will appear when conventional numerical simulations are used for solving the inverse problems. Thus, small perturbations added in the boundary conditions will result in problems of instability and large numerical errors. In this article, we adopt the GFDM to stably and accurately analyze the inverse problems governed by fourth-order partial differential equations. The GFDM is truly free from time-consuming mesh generation and numerical quadrature. Six numerical examples are provided to validate the accuracy and the simplicity of the GFDM. Furthermore, different levels of n...

Differential operators for a scale of Poisson type kernels in the unit disc

Abstract: In this paper, we introduce a scale of differential operators which is shown to correspond canonically to a certain scale of solution kernels generalizing the classical Poisson kernel for the unit disc. The scale of kernels studied is very natural and appears in many places in mathematical analysis, such as in the theory of integral representations of biharmonic functions in the unit disc.

On stable solutions of the biharmonic problem with polynomial growth

Abstract: We prove the nonexistence of smooth stable solution to the biharmonic problem � 2 u = u p , u > 0 in R N for 1 < p < 1 and N < 2(1 + x0), where x0 is the largest root of the following equation: x 4 32p(p + 1)

A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes

Abstract: The discrete reliability of a finite element method is a key ingredient to prove optimal convergence of an adaptive mesh-refinement strategy and requires the interchange of a coarse triangulation and some arbitrary refinement of it. One approach for this is the careful design of an intermediate triangulation with one-level refinements and with the remaining difficulty to design some interpolation operator which maps a possibly nonconforming approximation into the finite element space based on the finer triangulation. This paper enfolds the second possibility of some novel discrete Helmholtz decomposition for the nonconforming Morley finite element method. This guarantees the optimality of a standard adaptive mesh-refining algorithm for the biharmonic equation. Numerical examples illustrate the crucial dependence of the bulk parameter and the surprisingly short pre-asymptotic range of the adaptive Morley finite element method.

An efficient computational method for solving fractional biharmonic equation

Abstract: In this paper, we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval [0,L], generated by the shifted Chebyshev polynomials. Moreover, we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation. The main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem. Convergence of the shifted Chebyshev polynomials expansion in two-dimensions is investigated. Also the power of this manageable method is illustrated.

New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime

Abstract: In this work, we study energy of timelike biharmonic particle in a new spacetime Heisenberg spacetime $\mathcal {H}_{1}^{4}$ . We give a geometrical description of energy of a Frenet vector fields of timelike biharmonic particle in $\mathcal {H}_{1}^{4}.$ Moreover, we obtain different cases for this particles.

Multiple solutions of p-biharmonic equations with Navier boundary conditions

Abstract: In this article, exploiting variational methods, the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator is investigated. Moreover, a concrete example of an application is presented.

A Hybridized Weak Galerkin Finite Element Method for the Biharmonic Equation

Abstract: This paper presents a hybridized formulation for the weak Galerkin finite element method for the biharmonic equation. The hybridized weak Galerkin scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier is verified to provide a numerical approximation for certain derivatives of the exact solution. An optimal order error estimate is established for the numerical approximations arising from the hybridized weak Galerkin finite element method. The paper also derives a computational algorithm (Schur complement) by eliminating all the unknown variables on each element, yielding a significantly reduced system of linear equations for unknowns on the boundary of each element.

A bi-level view of inpainting - based image compression

Abstract: Inpainting based image compression ap- proaches, especially linear and non-linear diffusion models, are an active research topic for lossy image compression. The major challenge in these compression models is to find a small set of descriptive supporting points, which allow for an accurate reconstruction of the original image. It turns out in practice that this is a challenging problem even for the simplest Laplacian interpolation model. In this paper, we revisit the Laplacian interpolation compression model and introduce two fast algorithms, namely successive preconditioning primal dual algorithm and the recently proposed iPiano algorithm, to solve this problem efficiently. Furthermore, we extend the Laplacian interpolation based compression model to a more general form, which is based on principles from bi-level optimization. We investigate two different variants of the Laplacian model, namely biharmonic interpolation and smoothed Total Variation regularization. Our numerical results show that significant improvements can be obtained from the biharmonic inter- polation model, and it can recover an image with very high quality from only 5% pixels.

Shape deformation for vibrating hinged plates

Abstract: We consider the biharmonic operator subject to homogeneous intermediate boundary conditions of Steklov-type. We prove an analyticity result for the dependence of the eigenvalues upon domain perturbation and compute the appropriate Hadamard-type formulas for the shape derivatives. Finally, we prove that balls are critical domains for the symmetric functions of multiple eigenvalues subject to volume constraint. Copyright © 2013 John Wiley & Sons, Ltd.

Divergence-conforming Discontinuous Galerkin Methods and C^0 Interior Penalty Methods

Abstract: In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. Then, extensions of the concept to discontinuous Galerkin methods defined through lifting operators, for different weak formulations of the Stokes problem, and to three dimensions are discussed. Application of the equivalence result yields an optimal error estimate for the Stokes velocity without involving the pressure. Conversely, combined with a recent multigrid method for Stokes flow, we obtain a simple and uniform preconditioner for harmonic problems with simply supported and clamped boundary.

Vibrational higher-order resonances in an overdamped bistable system with biharmonic excitation

Abstract: Experimental evidence of vibrational higher-order resonances in a bistable vertical-cavity surface-emitting laser driven by two harmonic signals with very different frequencies is reported. The phenomenon shows up in a parameter space (the dc current, the amplitude of the high-frequency signal) as well-defined structures with multiple local maxima at higher harmonics of the low-frequency signal. Such structures appear due to a strong suppression of higher harmonics for certain values of the high-frequency amplitude and the dc current. Complexity of the structures and the total number of the local maxima depend on the harmonic order $k$. The behavior of nonlinear distortion factor is also studied. The experimental results are in a good agreement with the numerical results which were obtained in the model of the bistable overdamped oscillator with biharmonic excitation.

A Hybridized Weak Galerkin Finite Element Method for the Biharmonic Equation

Abstract: This paper presents a hybridized formulation for the weak Galerkin finite element method for the biharmonic equation. The hybridized weak Galerkin scheme is based on the use of a Lagrange multiplier defined on the element boundaries. The Lagrange multiplier is verified to provide a numerical approximation for certain derivatives of the exact solution. An optimal order error estimate is established for the numerical approximations arising from the hybridized weak Galerkin finite element method. The paper also derives a computational algorithm (Schur complement) by eliminating all the unknown variables on each element, yielding a significantly reduced system of linear equations for unknowns on the boundary of each element.

A C0-Weak Galerkin Finite Element Method for the Biharmonic Equation

Abstract: A C0-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for C0 functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete H2 norm and the standard H1 and L2 normswith appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined ScottZhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order (k + 1 − d) and the surface mass of order (k + 2− d) for the Pk+2 finite element functions in d-dimensional space.

A Fast Fourier--Galerkin Method Solving a Boundary Integral Equation for the Biharmonic Equation

Abstract: We develop a fast Fourier--Galerkin method for solving a boundary integral equation which is a reformulation of the Dirichlet problem of the biharmonic equation. The proposed method is based on a splitting of the resulting boundary integral operator. That is, we write the operator as a sum of two integral operators, one having the Fourier basis functions as eigenfunctions and the other whose representation matrix in the Fourier basis can be compressed to a sparse matrix having only $\mathcal{O}(n\log n)$ number of nonzero entries, where $n$ is the order of the Fourier basis functions used in the method. We then project the solution of the boundary integral equation onto the space spanned by the Fourier basis functions. This leads to a system of linear equations. A fast solver for the system is based on a compression of its coefficient matrix. We show that the method has the optimal convergence order $\mathcal{O}(n^{-q})$, where $q$ denotes the degree of regularity of the exact solution of the boundary int...

3D elastic solutions for laterally loaded discs: generalised Brazilian and Point Load tests

Abstract: This paper investigates the application of a double Fourier series technique to the construction of an elastic stress field in a cylindrical bar subject to lateral boundary loads. The lateral loads, including the constant load boundary conditions, are represented by two Fourier series: one on the perimeter of the circular section (r 0, θ) and the other on the longitudinal curved surface parallel to the bar axis (z). The technique invokes acceptable potential functions of the Papkovich–Neuber displacement field, satisfying the governing partial differential equations, to assign appropriate odd and even trigonometric Fourier terms in cylindrical coordinates (r, θ, z). The generic solution decomposes the problem of interest to a state of stress caused by two independent boundary conditions along the z axis and θ-polar angle, both superimposed on a solution for which these potentials are the product of the trigonometric terms of the independent variables (θ, z). Constants appearing in the resultant second-order partial differential equations are determined from the generally mixed (tractions and/or displacements) boundary conditions. While the solutions are satisfied exactly at the ends of an infinite bar, they are satisfied weakly on average, in the light of Saint Venant’s approximation at the two ends of a finite bar. The application of the proposed analysis is verified against available elastic solutions for axisymmetric and non-axisymmetric engineering problems such as the indirect Brazilian Tensile Strength and Point Load Strength tests.

Nonlinear Biharmonic Problems with Singular Potentials

Abstract: We deal with the problem \begin{eqnarray} \Delta^2 u +V(|x|)u = f(u), u\in D^{2,2}(R^N) \end{eqnarray} where $\Delta^2$ is biharmonic operator and the potential $V > 0 $ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of nontrivial radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions.