Search

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated.

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors.

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 H 2 -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.

There are three criteria when we model with phase-field equations. The first one is the evolution of phase separation. The second one is energy dissipation law. The last one is whether we have the law of conservation of mass or not. If we minimize the Ginzburg–Landau free-energy functional, the kinetics of phase separation inherently follows. However, the other properties such as energy dissipation and mass conservation are more difficult to model. Hitherto, there have been a few proposals to conserve mass using the Allen–Cahn (AC) equation, but we do not have the energy dissipation law in these models when considered in light of local effects of mass-correction. For the reasons, certain researchers still prefer to use the Cahn–Hilliard equation, e.g. biharmonic equation instead of using the Allen–Cahn equation when we consider a mass conserving case. This paper is concerned with numerical solutions of the AC equation in the mass-conserving space, where the solution is corrected at each iteration for mass conservation. We introduce two different methods for solving the conservative AC equation with periodic boundary conditions. In this work, we have the AC equation in common, but use two different mass correction operators. Note that both of mass correction operators satisfy the discrete energy dissipation property. First, we project the solution of the AC equation to the mass conserving space, and then recover the linear convex splitting scheme for the conventional mass-conserving equation. The linear convex splitting scheme is well-known but the convergence of numerical solutions is elaborately studied in terms of gradient-projection. Next, we propose the other energy-minimizing method on the mass-conserving space. It translates the solution of the AC equation to the mass conserving space via an energy-dissipation operator. The dissipation of discrete energy functional for the proposed method is also established. As far as the author knows, no report has been presented for the numerical scheme having discrete energy dissipation property of the mass-conserving AC equation without using the nonlocal and constant multiplier. To verify the common properties of mass-projection and proposed methods numerically, we simulate the evolutions for phase separation, dissipation of energy, and mass conservation. We moreover consider the short and long time stages of evolutions to see the difference between two methods. These several tests are provided for its applications of the proposed method for modeling natural phenomena.

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function ψ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable ψ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

In this paper, we are devoted to the study of critical p -biharmonic system with a parameter λ , which involves strongly-coupled critical nonlinearities and negative exponents. We prove that there exists a positive constant λ ∗ such that the above problem admits at least two solutions if λ ∈ ( 0 , λ ∗ ) .

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, H 2 -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 h p -adaptivity recover optimal algebraic and exponential converge rates, respectively.

Using the idea of differential quadrature, two new methods are constructed to approximate the solution of elliptic partial differential equations in higher dimensions. To obtain the weighting coefficients of the first method, a mixture of grid points and mid points of the uniform partition is used. The order of convergence obtained by the first algorithm is non-optimal, so we mixed the idea with spline collocation to obtain higher order approximations. Using the weighting coefficients of the non-optimal algorithm, some new weighting coefficients are obtained which are used to obtain higher accuracy. For the first time, a sixth order approximation is obtained to the solution of well-known high order multi-dimensional elliptic PDEs such as biharmonic and triharmonic problems. We found a way to increase the order of convergence and the accuracy without increasing the CPU runtime. The block form of the coefficients matrix for the linear case is presented to have a better vision on the system arised from the method. Some examples of biharmonic and triharmonic problems are solved to show the good performance and applicability of the proposed algorithms.

In this paper, for investigating a boundary value problem for a nonlinear biharmonic equation of Kirchhoff type, we propose an efficient approach by the reduction of the problem to an operator equation for the nonlinear part of the equation. The result is that we have established the existence and uniqueness of a solution under some easily verified conditions. Also, we propose an iterative method for finding the solution. Some examples demonstrate the applicability of the theoretical results and the efficiency of the iterative method.

In this paper, an upwind compact difference method with second-order accuracy both in space and time is proposed for the streamfunction–velocity formulation of the unsteady incompressible Navier–Stokes equations. The first derivatives of streamfunction (velocities) are discretized by two type compact schemes, viz. the third-order upwind compact schemes suggested with the characteristic of low dispersion error are used for the advection terms and the fourth-order symmetric compact scheme is employed for the biharmonic term. As a result, a five point constant coefficient second-order compact scheme is established, in which the computational stencils for streamfunction only require grid values at five points at both ( n ) th and ( n + 1 ) th time levels. The new scheme can suppress non-physical oscillations. Moreover, the unconditional stability of the scheme for the linear model is proved by means of the discrete von Neumann analysis. Four numerical experiments involving a test problem with the analytic solution, doubly periodic double shear layer flow problem, lid driven square cavity flow problem and two-sided non-facing lid driven square cavity flow problem are solved numerically to demonstrate the accuracy and efficiency of the newly proposed scheme. The present scheme not only shows the good numerical performance for the problems with sharp gradients, but also proves more effective than the existing second-order compact scheme of the streamfunction–velocity formulation in the aspect of computational cost.

The idea of differential quadrature is used to construct a new algorithm for the solution of differential equations. To determine the weighting coefficients of DQM, B-spline basis functions of degree r are used as test functions. The method is constructed on a set of points mixed from grid points and mid points of a uniform partition. Using the definition of B-splines interpolation as alternative, some error bounds are obtained for DQM. The method is successfully implemented on nonlinear boundary value problems of order m. Also the application of the proposed method to approximate the solution of multi-dimensional elliptic PDEs is included in the paper. As test problem, some examples of biharmonic and Poisson equations are solved in 2D and 3D. The results are compared with some existing methods to show the efficiency and performance of the proposed algorithm. Also some examples of time dependent PDEs are solved to compare the results with other existing spline based DQ methods.

In the present paper, we consider the following modified quasilinear fourth-order elliptic equation (1.1) { Δ 2 u − Δ u + V ( x ) u − 1 2 Δ ( u 2 ) u = f ( x , u ) , for x ∈ R N , u ( x ) ∈ H 2 ( R N ) , where Δ 2 : = Δ ( Δ ) is the biharmonic operator, V ∈ C ( R N , R ) and f ∈ C ( R N × R , R ) , N ≤ 5 , are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods.

The study of fourth order partial differential equations has flourished in the last years, however, a p()-biharmonic problem with no-flux boundary condition has never been considered before, not even for constant p. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in order to establish the multiplicity result, we do not impose an AmbrosettiRabinowitz type condition, nor a symmetry condition, on our nonlinearity f.

We study the linear space of C s -smooth isogeometric functions defined on a multi-patch domain? ? ? R 2 . We show that the construction of these functions is closely related to the concept of geometric continuity of surfaces, which has originated in geometric design. More precisely, the C s -smoothness of isogeometric functions is found to be equivalent to geometric smoothness of the same order ( G s -smoothness) of their graph surfaces. This motivates us to call them C s -smooth geometrically continuous isogeometric functions. We present a general framework to construct a basis and explore potential applications in isogeometric analysis. The space of C 1 -smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains is analyzed in more detail. Numerical experiments with bicubic and biquartic functions for performing L 2 approximation and for solving Poisson's equation and the biharmonic equation on two-patch geometries are presented and indicate optimal rates of convergence.

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., Comment: 25 pages. arXiv admin note: text overlap with arXiv:1303.0927

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.

In this paper, a C-0 interior penalty method has been proposed and analyzed for distributed optimal control problems governed by the biharmonic operator. The state and adjoint variables are discretized using continuous piecewise quadratic finite elements while the control variable is discretized using piecewise constant approximations. A priori and a posteriori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions. Numerical results justify the theoretical results obtained. The a posteriori error estimators are useful in adaptive finite element approximation and the numerical results indicate that the sharp error estimators work efficiently in guiding the mesh refinement. (C) 2014 Elsevier Ltd. All rights reserved.

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.

In order to explore a new approach to construct surfaces bounded by geodesics or lines of curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First, the corresponding biharmonic surface can naturally be derived under some degenerative conditions; second, the parameters in the PDE implicate some physical meaning, such as elasticity or rigidity; third, there are only a few parameters that need to be evaluated, and hence the computation is simple. In addition, this paper constructs two adjacent surfaces with C^1 continuity whose common boundary is the same given curve as well as respective geodesic (or line of curvature). Examples show that this method to construct PDE-based surfaces bounded by geodesics or lines of curvature is easy and effective.

The biharmonic equivalent of Poisson's equation is defined here as follows. The biharmonic operator on potential at a point density is double the right-hand side of Poisson's equation for the same point density. The mathematical proof of this and other matters are provided in the body of this presentation. This relationship has brought about the replacement of the point mass model with a point density model as the first step of the procedure. The point density model avoids the difficulties associated with point masses. Point masses produce impulse responses at zero distances. Point densities produce null responses at zero distances. Our point density model works extremely well for interpolating scattered geodetic surface data. The unknowns in our system of equations are interpretable as point density anomalies. It is customary to place the point density anomalies at shallow or near zero depths in the vicinity of the surface data. This cannot be done with points masses. We have not been concerned with the possible usefulness of the geodetically estimated point densities for other purposes. They simply provide the basis for smooth interpolation of several geodetic parameters, which fit data values exactly. In any case, we are not competing with seismologists who in a sense are measuring densities, instead of estimating them. In fact, we believe that geodetic interpolation could be improved with the use of seismic density data. That is why we are proposing a joint venture in seismic/geodetic subsurface analysis.

The domain decomposition method proposed by Schwarz [1] is gaining significance in view of the possibilities for parallel implementation. In this paper, we apply the domain decomposition method to the biharmonic equation in two overlapping discs. In each subdomain, a boundary integral formula is used to obtain the solution. Unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain, the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries. Another aspect of the paper is that, in order to handle the operations with dense matrices arising from the boundary integral formulation, the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy. Examples involving a Laplace equation in an L-shaped region and a domain bounded by a cardioid and a circle are also illustrated.

A spectral Galerkin scheme for solving the two-dimensional stationary Stokes problem is considered. In vorticity-stream function formulation, the Stokes problem is reduced to a biharmonic one. A new formulation for the decomposition of the biharmonic operator is introduced. Main difficulty in this decomposition is to find the trace of the vorticity. Using the adjoint of the composition of trace operator and Green's operator, a variational form is obtained for computing the boundary value of the vorticity. The discrete variational form preserves the properties such as coercivity and continuity of the continuous case which are important for our convergence analysis. Convergence of our scheme is proved, and numerical experiments are performed for the steady driven cavity problem.

A numerical integral-equation method is presented for the accurate solution of viscous flow problems in which there are inherent boundary singularities. Where applicable, this method has the advantage over existing singularity subtraction and singularity incorporation methods since natural boundary conditions can be employed without any special form of pre- or post-processing. Results are presented for the benchmark problem of Stokes flow in a lid-driven cavity.

The potential theoretic foundations proposed by Hardy for his multiquadric-biharmonic method are shown to be either erroneous or deficient for the purpose of gravity predictions based on mixed data types. Specifically, the discretized model for the potential as a sum of point sources weighted by distance is nonharmonic in free space; whereas, the relationships between gravitational quantities (gravity anomaly, geoid undulation, deflections of the vertical, etc.) are predicated on a Newtonian (harmonic) potential.

An algorithm for the solution of the finite difference approximation of the perturbed biharmonic problem is proposed which consists of both direct and iterative stages. The optimality of this algorithm is proved under the opportune hypothesis on the perturbation.

A knowledge of the complex roots λ of the transcendental eigenvalue equation sin λα =± λ sin α is essential in the analysis of the slow viscous fluid flow in the neighbourhood of a sharp corner which subtends an angle α ϵ (0, 2π] to the fluid. Existing methods for finding all roots λ essentially require an a priori knowledge of the solution structure; given that { λ 1 , λ 2 , …, λ m } are known, gl m +1 is determined via iterations, and/or a solution procedure initiated by λ m . We present herein a general interval analysis method which exhaustively finds all roots λ via only the original eigenvalue equation: no other information is required. The interval-analysis method automatically guarantees root existence and uniqueness while simultaneously providing error bounds.

In this paper, new numerical algorithms of finite element methods (FEM) are reported for both biharmonic equations and 3D blending surfaces, to achieve the global superconvergence O(h3)-O(h4) in H2 norms. This is significant, compared with the optimal convergence O(h2). The algorithms are simple because only an a posteriori interpolant solution is needed. Such a global convergence method was originated by Lin and his colleagues in [1–3] for only the clamped boundary conditions. Recently, we extended the global superconvergence to other boundary conditions, such as the simple support condition, the periodic boundary, and the natural boundary condition. Moreover, we apply in [4–6] this global superconvergence to the FEM using the penalty techniques for biharmonic equations and blending problems, also to reach O(h3) and O(h4) for quasiuniform and uniform □ij, respectively. Currently, we develop in [7,8] and in this paper the FEM using the penalty plus hybrid techniques to reduce the condition number down to O(h−4)-O(h−5) of the associated matrix, while retaining superconvergence O(h3)-O(h4). Since instability is severe for biharmonic equations, any reduction of the condition number is crucial. By the new algorithms in this paper, not only can a great deal of CPU time be saved, but also the complicated biharmonic equations and blending surfaces may be solved in double precision. Numerical experiments are carried out to support the theoretical conclusions.

In this paper, it is proved that the two approaches, known in the literature as the method of fundamental solutions (MFS) and the Trefftz method, are mathematically equivalent in spite of their essentially minor and apparent differences in formulation. In deriving the equivalence of the Trefftz method and the MFS for the Laplace and biharmonic problems, it is interesting to find that the complete set in the Trefftz method for the Laplace and biharmonic problems are embedded in the degenerate kernels of the MFS. The degenerate scale appears using the MFS when the geometrical matrix is singular. The occurring mechanism of the degenerate scale in the MFS is also studied by using circulant. The comparison of accuracy and efficiency of the two methods was addressed.

This paper describes the finite element approximations based on a stationary variational principle of the biharmonic equation. Independent approximations for the solution and its second derivatives are used in the element: the normal derivatives of the solution and its second derivatives are approximated independently on the element boundary. Thus, the present model is a ‘mixed-hybrid’ model. Existence and uniqueness of solutions to the exact weak (or variational) problem are established and the associated finite element approximations are described. Existence and uniqueness of the finite element solutions are proved and a priori error estimates are given.

We develop local mode analysis of multicolor (or more generally multicoupling) numerical processes and provide practical procedures for computing their symbols and predicting their convergence and smoothing properties, in the context of multigrid methods. We apply our analysis, including simple numerical examples, to multicolor point relaxation, especially in the context of the novel composite relaxation (A. Brandt, Barriers to achieving textbook multigrid efficiency (TME) in CFD, Tech. Report, Gauss Center Report WI/GC-10, Weizmann Institute, December 1998 ; O. E. Livne, Multiscale Eigenbasis Algorithms, Ph.D. Thesis, The Weizmann Institute of Science, Rehovot, Israel, 2000 ) for products of discrete operators, such as the biharmonic.