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We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]

We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally C 4 -smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L 2 -norm as well as to equivalents of the H s -seminorms for 1 ≤ s ≤ 4. In the studied case of spline degree p = 9 , the numerical results indicate in case of the Greville points a convergence of order O (h p − 3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O (h p − 2) for all (semi)norms except for the equivalent of the H 4 -seminorm, where the order O (h p − 3) is anyway optimal. [ABSTRACT FROM AUTHOR]

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted P k + 4 polynomial from four P k polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost- C 1 , Analysis-Suitable G 1 and the Approximate C 1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C 1 and Analysis-Suitable G 1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost- C 1 and D-Patch provide relatively easy construction on complex geometries. The Almost- C 1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C 1 and Analysis-Suitable G 1 applicable to more complex geometries. [ABSTRACT FROM AUTHOR]

• Violation of the Saint–Venant compatibility condition is considered. • Equation for the Airy stress function in the non-Euclidean model is obtained. • The Airy stress function is calculated through the incompatibility function. • Residual stress are determined in the framework of the non-Euclidean model. We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions. [ABSTRACT FROM AUTHOR]

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems. [ABSTRACT FROM AUTHOR]

• A novel scheme is developed to simulate the thin plate bending problems. • The proposed method is meshless without numerical integration and iteration. • The presented method is quite accurate. This study develops a simple and effective numerical technique, which aims to accurately and quickly address thin plate bending problems. Based on the given boundary conditions, the thin plate homogenization function is constructed and a family of two-parameter homogenization functions are derived. Then, the superposition of homogenization functions method for the thin plate, the clamped plate, and the simply supported plate is obtained, which is meshless without numerical integration and iteration with the merits of easy-to-program and easy-to-implement. Six numerical experiments are employed to verify the effectiveness, accuracy and convergence of the proposed novel strategy. The proposed method is evaluated by the comparisons with the analytical solutions and the referenced solutions. It can be observed that the proposed method is quite accurate for the thin plate, the clamped plate, and the simply supported plate problems. [ABSTRACT FROM AUTHOR]

We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H 2 norm and a L 2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation. [ABSTRACT FROM AUTHOR]

For g ∈ C (D ‾) , φ ∈ C (T) and f ⁎ ∈ C (T) , let BH φ , g , f ⁎ (D) be the class of all complex-valued functions f ∈ C 4 (D) satisfying Δ (Δ f) = g in the unit disk D with Δ f = φ in the unit circle T and f = f ⁎ in T , where Δ f is the Laplacian of f. In this article, we will show that several Schwarz type lemmas for f ∈ BH φ , g , f ⁎ (D) , and, by applying these results, we will establish a Landau type theorem on a subclass of BH φ , g , f ⁎ (D). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the adaptive fashion of the Ciarlet–Raviart mixed method for biharmonic equation/eigenvalue problem with simply supported boundary condition in R d. We propose an a posteriori error indicator of the Ciarlet–Raviart approximate solution for the biharmonic equation and an a posteriori error indicator of the Ciarlet–Raviart approximate eigenfuction, and prove the reliability and efficiency of the indicators. We also give an a posteriori error indicator for the approximate eigenvalue and prove its reliability. We design an adaptive Ciarlet–Raviart mixed method with piecewise polynomials of degree less than or equal to m , and numerical experiments show that numerical eigenvalues obtained by the method can achieve the optimal convergence order O (d o f − 2 m d ) (d = 2 , m = 2 , 3 ; d = 3 , m = 3). [ABSTRACT FROM AUTHOR]

In this paper, we analyze the existence of positive solution to a biharmonic equation with Navier boundary conditions, and formulate simple uniqueness conditions for the positive solution. In addition, a necessary condition for the existence of the positive solution is also derived. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We present and analyze an interior penalty discontinuous Galerkin Isogeometric Analysis (dG-IgA) method for the biharmonic equation in computational domain in R d with d = 2 , 3 . The computational domain consists of several non-overlapping sub-domains or patches. We construct B-Spline approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate in a discrete norm and numerical experiments to confirm the theory. [ABSTRACT FROM AUTHOR]

Finite pointset method is one of the grid free methods that is used to solve differential equations arising from physical problems. It is a local iterative procedure based on weighted least square approximation technique. In this paper, biharmonic equation with simply supported, clamped and Cahn–Hilliard type boundary conditions, is solved using the finite pointset method. Numerical examples illustrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

By introducing a subspace of H 2 ( Ω ) with constraints ∂ u ∂ n | ∂ Ω = 0 and ∫ Ω u d x = 0 and using the Fountain Theorem, we obtain the existence of infinitely many sign-changing high energy solutions for a biharmonic equations with p -Laplacian and Neumann boundary condition. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We generate a basis of the space of bicubic and biquartic C 1 -smooth geometrically continuous isogeometric functions on bilinear multi-patch domains Ω ⊂ R 2 . The basis functions are obtained by suitably combining C 1 -smooth geometrically continuous isogeometric functions on bilinearly parameterized two-patch domains (cf. [18] ). They are described by simple explicit formulas for their spline coefficients. These C 1 -smooth isogeometric functions possess potential for applications in isogeometric analysis, which is demonstrated by several examples (such as the biharmonic equation). In particular, the numerical results indicate optimal approximation power. [ABSTRACT FROM AUTHOR]

Here we give a self-contained new proof of the existence of singular solutions to a class Dirichlet problem with a singular nonlinearity. These results were obtained by Dávila–Flores–Guerra (Dávila et al., 2009). Our proof is much more direct and simpler. [ABSTRACT FROM AUTHOR]

A two-dimensional analytical solution is presented for stresses and displacements in an elastic curved beam forming an incomplete ring in frictionless and unbonded contact with a rigid pin loaded by a point force and in the presence of clearance. The circular beam is modeled as an incomplete elastic thick ring, constrained at both ends and in a plane stress state. The stress and displacement fields within the beam are derived from a biharmonic Airy stress function, according to the Michell solution in polar coordinates. The mixed boundary value problem is reduced to a set of dual series equations and then to a non-homogeneous linear system of infinite equations, which is then solved by truncation. The non-linear relations between the applied load and the contact angle or the pressure distribution are obtained by using an inverse method. The analytical results are compared with finite element predictions for a pin-lug connection and a reasonable agreement is observed for several typical geometries. The peaks of contact pressure and von Mises equivalent stress and their location within the curved beam are evidenced. [ABSTRACT FROM AUTHOR]

We study the quasilinear biharmonic problem { Δ ( | Δ u | p − 2 Δ u ) = f ( | x | , u ) + g ( | x | ) , x ∈ B 1 , u = Δ u = 0 , x ∈ ∂ B 1 , where B 1 = { x ∈ R N : | x | < 1 } with N ≥ 2 . Criteria are established for the existence of at least one positive radial solution of the problem. Explicit lower and upper bounds of the solutions are obtained in our results. [ABSTRACT FROM AUTHOR]

We present an approximately C 1 -smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [1] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C 0. Therefore, to obtain C 1 -smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff–Love plate/shell formulations, using an isogeometric Galerkin method. Isogeometric spaces that are globally C 1 over multi-patch domains can be constructed as in [2–6]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G 1 parametrizations. To allow C 1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C 1 conditions. Thus, we define function spaces that are not exactly C 1 but only approximately. We adopt the construction for two-patch domains, as developed in [1] , and extend it to more general multi-patch domains. We employ the construction for a biharmonic model problem and compare the results with Nitsche's method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h -refinement, comparable to the solution using Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate C 1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated. [ABSTRACT FROM AUTHOR]

We prove the convergence of an innovative Chebyshev–Gauss–Lobatto (CGL) pseudospectral method applied to fourth order boundary value problems. The proposed method enjoyed all the advantages of pseudospectral methods. Moreover, we can select (N − 3) interior CGL collocation points to enforce the equation, meanwhile we use the remaining four collocation points to assure the boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the entire solution of the semilinear biharmonic equation Δ 2 u = e u , in R N , N ≥ 1 . For N = 3 , we obtain the asymptotic of the entire nonradial solution which extends the results of Lai and Ye (in press). Besides, a new singular solution of the above equation is constructed by derivative estimates for N ≥ 13 . [ABSTRACT FROM AUTHOR]

We study the linear space of C s -smooth isogeometric functions defined on a multi-patch domain Ω ⊂ 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. [ABSTRACT FROM AUTHOR]

Meshless method choosing Heaviside step function as a test function for solving simply supported thin plates under various loads is presented in this paper. The shape functions using regular and irregular nodal distribution as well as order of polynomial basis choice are constructed by moving kriging interpolation. Alternatively, two-field-variable local weak forms are used in order to decompose the governing equation, biharmonic equation, into a couple of Poisson equations and then impose straightforward boundary conditions. Selected numerical examples are considered to examine the applicability, the easiness, and the accuracy of the proposed method. Comparing to an exact solution, this robust method gives significantly accurate numerical results, implementing by maximum relative error and root mean square relative error. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: The classical a posteriori error estimates are mostly oriented to the use in the finite element -methods while the contemporary higher-order -methods usually require new approaches in a posteriori error estimation. These methods hold a very important position among adaptive numerical procedures for solving ordinary as well as partial differential equations arising from various technical applications. In the paper, we are concerned with a review and comparison of error estimation procedures for the biharmonic and some more general fourth order partial differential problems with special regards to the needs of the -method. We point out some advantages and drawbacks of analytical and computational a posteriori error estimates. [Copyright &y& Elsevier]

Abstract: In this paper, we present a computational method for solving 2D and 3D Poisson equations and biharmonic equations which based on the use of Haar wavelets. The highest derivative appearing in the differential equation is expanded into the Haar series, this approximation is integrated while the boundary conditions are incorporated by using integration constants. In 2D the first transform the spectral coefficients into the nodal variable values and then use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solutions to four test problems are investigated. [Copyright &y& Elsevier]

Abstract: The prevalence of micromoulded components has steadily increased over recent years. The production of such components is extremely sensitive to a number of variables that may potentially lead to significant changes in the surface geometry, often regarded as a crucial determinant of the product’s functionality and quality. So far, traditional large-scale quality assessment techniques have been used in micromoulding. However, these techniques are not entirely suitable for small scales. Techniques such as atomic force microscopy (AFM) or white light interferometry (WLI) have been used for obtaining full three-dimensional profiles of micromoulded components, producing large data sets that are very difficult to manage. This work presents a method of characterizing surface features of micro and nano scale based on the use of the Biharmonic equation as means of describing surface profiles whilst guaranteeing tangential (C 1) continuity. Thus, the problem of representing surface features of micromoulded components from massive point clouds is transformed into a boundary-value problem, reducing the amount of data required to describe any given surface feature. The boundary conditions needed for finding a particular solution to the Biharmonic equation are extracted from the data set and the coefficients associated with a suitable analytic solution are used to describe key design parameters or geometric properties of a surface feature. Moreover, the expressions found for describing key design parameters in terms of the analytic solution to the Biharmonic equation may lead to a more suitable quality assessment technique for micromoulding than the criteria currently used. In summary this technique provides a means for compressing point clouds representing surface features whilst providing an analytic description of such features. The work is applicable to many other instances where surface topography is in need of efficient representation. [Copyright &y& Elsevier]

Abstract: In this paper we study the critical growth biharmonic problem with a parameter and establish uniform lower bounds for , which is the supremum of the set of , related to the existence of positive solutions of the biharmonic problem. [Copyright &y& Elsevier]

Abstract: By a variant version of Fountain Theorem due to Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms. [Copyright &y& Elsevier]

Abstract: In this work, we present a computational method for solving Poisson equations and biharmonic equations which are based on the use of Haar wavelets. The first transform the spectral coefficients into the nodal variable values. The second use Kronecker products to construct the approximations for derivatives over a tensor product grid of the horizontal and vertical blocks. Finally, solve the obtained system of algebraic equations. The efficiency of the method is demonstrated by four numerical examples. [Copyright &y& Elsevier]

Abstract: This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches. [Copyright &y& Elsevier]

Abstract: We propose a stabilized finite element method for the approximation of the biharmonic equation with a clamped boundary condition. The mixed formulation of the biharmonic equation is obtained by introducing the gradient of the solution and a Lagrange multiplier as new unknowns. Working with a pair of bases forming a biorthogonal system, we can easily eliminate the gradient of the solution and the Lagrange multiplier from the saddle point system leading to a positive definite formulation. Using a superconvergence property of a gradient recovery operator, we prove an optimal a priori estimate for the finite element discretization for a class of meshes. [Copyright &y& Elsevier]