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The plate bending problems in the Kirchhoff–Love model can be reduced to biharmonic equations. In the previous papers (Kim et al., 2020; Kim et al., 2022), we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ 2 u = f in a non-convex (cracked, L -shaped) domain with various boundary conditions such as clamped, simply connected, free, and mixed. In previous papers, we could not show a general SIF formula related to Poisson's ratio ν contained in the free boundary condition. Moreover, the previous SIF extraction formula obtained for a given Poisson's ratio requires cumbersome line integrals. In this paper, removing the line integrals in the extraction formulas, we show improved SIF extraction formulas applicable to biharmonic equations with free boundary conditions. Thus, the new, improved SIF extraction formulas for free boundary conditions become more straightforward and can be applied to plate bending problems. [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]

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]

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]

We propose two efficient, non-iterative, and second-order accurate algorithms to solving the anisotropic Allen–Cahn equation with the nonlocal mass conservation. The first scheme is the stabilized-SAV approach which combines the SAV approach with the stabilization technique, in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps. The scheme not only can conserve the mass precisely but also is very easy-to-implement and non-iterative where one only needs to solve three decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the scheme is unconditionally energy stable rigorously. The second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way, therefore the scheme is quite efficient and stable that allows for large time steps in computations. For both schemes, we present a number of 2D and 3D numerical simulations to show stability and accuracy. • The anisotropic Cahn-Hilliard model is reformulated by using nonlocal Allen-Cahn equation. • Two efficient, non-iterative, and second-order algorithms are developed to preserve the mass. • Three linear stabilization terms are added to enhance the stability that allows large time step. • Numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy. [ABSTRACT FROM AUTHOR]

We develop efficient and accurate numerical methods to solve a class of shallow shell problems of the von Karman type. The governing equations form a fourth-order coupled system of nonlinear biharmonic equations for the transverse deflection and Airy's stress function. A second-order finite difference discretization with three iterative methods (Picard, Newton and Trust-Region Dogleg) is proposed for the numerical solution of the nonlinear PDE system. Three simple boundary conditions and two application-motivated mixed boundary conditions are considered. Along with the nonlinearity of the system, boundary singularities that appear when mixed boundary conditions are specified are the main numerical challenges. Two approaches that use either a transition function or local corrections are developed to deal with these boundary singularities. All the proposed numerical methods are validated using carefully designed numerical tests, where expected orders of accuracy and rates of convergence are observed. A rough run-time performance comparison is also conducted to illustrate the efficiency of our methods. As an application of the methods, a snap-through thermal buckling problem is considered. The critical thermal loads of shell buckling with various boundary conditions are numerically calculated, and snap-through bifurcation curves are also obtained using our numerical methods together with a pseudo-arclength continuation method. Our results are consistent with previous studies. [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]

An energy space based Dirichlet boundary control problem governed by biharmonic equation is investigated and subsequently a C 0 -interior penalty method is proposed and analyzed. An abstract a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution is sufficiently regular. Further an optimal order L 2 -norm error estimate is derived. Numerical experiments illustrate the theoretical findings. [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]

Piecewise biquadratic B-Spline surface satisfying biharmonic condition is studied in this paper. By applying biharmonic PDE, biquadratic B-Spline surface is fully determined by boundary control vertices. A linear system for solving inner control vertices is established. Its coefficient matrix is block tridiagonal and the proof of non-singularity of coefficient matrix is presented under the condition that all knots do not coincide with each other. A few examples are given to show the effectiveness of piecewise biharmonic biquadratic B-Spline surface. An interpolation method on given boundary points is presented if boundary curves are open. [ABSTRACT FROM AUTHOR]

In this work, based on a finite difference scheme, we propose the weak Galerkin (WG) method for solving time-fractional biharmonic equations. Theoretically and numerically, the optimal error estimates for semi-discrete and fully discrete schemes have been investigated. Based on mathematical induction, stability is discussed for the fully discrete scheme that depends on the initial value and the source term. Numerical experiments are provided to confirm the theoretical claims made by the proposed schemes. • We introduced a flexible discontinuous numerical method. • We used the concept of weak functions and their weak derivatives. • A semi-discrete scheme based on (P1 , P1 , P1) and a fully-discrete scheme based on the finite difference method are investigated. [ABSTRACT FROM AUTHOR]

In this work, for the phase-field model of lipid vesicles with the property of accurate volume conservation, we construct an effective fully-discrete numerical scheme, in which, the time marching method is based on a novel splitting type technique, and space is discretized by using the Spectral-Galerkin method. The advantage of this scheme is its high efficiency and ease of implementation. Specifically, although the model is highly nonlinear, just by solving two independent linear biharmonic equations with constant coefficients at each time step, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. The essence of the scheme is to introduce several additional auxiliary variables and use the specially designed ordinary differential equations to reformulate the system. In this way, energy stability can be obtained unconditionally, while avoiding the calculation of variable-coefficient systems. We strictly prove that the energy stability in the fully-discrete form that the scheme holds and give a detailed implementation process. Numerical experiments in 2D and 3D are further carried out to verify the convergence rate, energy stability, and effectiveness of the developed algorithm. • For phase-field vesicle model, a new fully-discrete spectral-Galerkin scheme is designed. • The scheme is linear, spectral order of spatial accuracy, second-order temporal accuracy. • It only needs to solve two independent linear biharmonic equations with constant coefficients. • The unconditional energy stability rigorously is shown and various numerical simulations are given. [ABSTRACT FROM AUTHOR]

Abstract: The modified mapping-collocation (MMC) method was applied to the boundary value problem (BVP) of a circumferential crack in an isotropic elastic curved beam subjected to pure bending moment loading. The stress correlation technique is then used to determine opening and sliding mode stress intensity factor (SIF) values based on the computed stress field near the crack tip. The MMC method aims at solving two-dimensional BVP of linear elastic fracture mechanics (LEFM) circumventing the need for direct treatment of the biharmonic equation by combining the power of analytic tools of complex analysis (Muskhelishvili formulation, conformal mapping, and continuation arguments) with simplicity of applying the boundary collocation method as a numerical solution approach. A good qualitative agreement between the computed stress contours and the fringe shapes obtained from the photoelastic experiment on a plexiglass specimen is observed. A quantitative comparison with FEM results is also made using ANSYS. The effect of crack size, crack position and beam thickness variation on SIF values and mode-mixity is investigated. [Copyright &y& Elsevier]

In this paper, we present the theoretical analysis of the optimal order convergence for the piecewise linear and continuous finite element method based on the Ciarlet–Raviart mixed formulation of the biharmonic eigenvalue problem associated to the clamped boundary condition. As far as we know, only the convergence of the equal order linear Ciarlet–Raviart finite element method for the eigenvalue problem has been established on convex domains. The aim to this work is to derive the convergence under the minimum regularity requirement and prove an improved convergence rate for the approximate eigenvalues. We introduce the corresponding solution spaces naturally attached to the continuous and discrete problems and prove the spectral approximation and error estimate of the discrete scheme. Some numerical examples are shown for the validation of the theoretical proof. [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: A higher order numerical discretization technique based on Minimum Sobolev Norm (MSN) interpolation was introduced in our previous work. In this article, the discretization technique is presented as a tool to solve two hard classes of PDEs, namely, the exterior Laplace problem and the biharmonic problem. The exterior Laplace problem is compactified and the resultant near singular PDE is solved using this technique. This finite difference type method is then used to discretize and solve biharmonic type PDEs. A simple book keeping trick of using Ghost points is used to obtain a perfectly constrained discrete system. Numerical results such as discretization error, condition number estimate, and solution error are presented. For both classes of PDEs, variable coefficient examples on complicated geometries and irregular grids are considered. The method is seen to have high order of convergence in all these cases through numerical evidence. Perhaps for the first time, such a systematic higher order procedure for irregular grids and variable coefficient cases is now available. Though not discussed in the paper, the idea seems to be easily generalizable to finite element type techniques as well. [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]

Abstract: In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity. [Copyright &y& Elsevier]

Abstract: We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given. [ABSTRACT FROM AUTHOR]

Abstract: This paper shows that for any suitably smooth function and arbitrarily selected interpolation nodes in , the Filon method and the Levin method for with the polynomial interpolation approach are identical when is a linear function. Based on this result, a new efficient Levin quadrature for is presented. [Copyright &y& Elsevier]

Abstract: The rate of H-convergence of truncations of stochastic infinite-dimensional systemswith nonrandom, local Lipschitz-continuous operators and G acting on a separable Hilbert space H, where () is studied. For this purpose, some new kind of monotonicity conditions on those operators and an existing H-series expansion of the Wiener process W are exploited. The rate of convergence is expressed in terms of the converging series-remainder , where are the eigenvalues of the covariance operator Q of W. An application to the approximation of semilinear stochastic partial differential equations with cubic-type of nonlinearity is given too. [Copyright &y& Elsevier]

Abstract: We prove sharp embeddings of Besov spaces involving both the classical smoothness and a slowly varying smoothness b into Lorentz–Karamata spaces. Our methods are quite elementary, we use neither the interpolation theory nor the atomic decomposition of spaces in question. We cover both sub-limiting and limiting cases and we determine growth envelopes of spaces . [Copyright &y& Elsevier]

Abstract: The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He''s variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. [Copyright &y& Elsevier]

This paper presents a mixed finite volume element scheme based on rectangular partition for solving biharmonic equations. It also gives a kind of adaptive Uzawa iteration method for the scheme. It is rigorously proved that the scheme has first-order accuracy in H1 semi-norm and L2 norm according to the characteristics of the scheme. Finally, two numerical examples illustrate the effectiveness of the method. [Copyright &y& Elsevier]

For the global superconvergence over the entire solution domain originated by Lin and his colleagues [7,8], this paper gives a framework of new estimates and new proofs for the basic estimates for bounds of ∫∫lower limit Ω (u−uI)xxvxx ds, ∫∫lower limit Ω (u−uI)xyvxy ds and ∫∫lower limit Ω (u−uI)xxvyy ds, which reveal more intrinsic characteristics and easier understanding and better readable. Suppose that the solution is smooth enough and the solution domain can be split into quasiuniform rectangular elements □ij with the maximal boundary length h. The study of [7,8] dealt with only the clamped boundary condition for biharmonic equations, to obtain the global superconvergence O(h4) in H2 norms under uniform rectangles □ij for the solution u∈H6(Ω). This paper is devoted to other kinds of important boundary conditions, such as the simple support condition, the natural boundary condition and their mixed types where the different boundary conditions are subject to different edges of ∂Ω. New error estimates are derived theoretically, and verified numerically to reach the global superconvergence O(h3.5) and O(h4) for different boundary conditions on different edges of Ω under uniform □ij. Note that the new superconvergence estimates in this paper are essential in practical applications, because different boundary conditions are needed in 3D blending surfaces [5] and in the combined methods for singularity problems [6]. [Copyright &y& Elsevier]

In addition to the various uses it was introduced for, the theory of Γ-convergence offers a rather natural setting for discussing and developing nonorthodox approximation methods for variational problems. For certain boundary value problems involving the bi-Laplacian, sequences of discrete functionals are here defined and are shown to Γ-converge to the corresponding functionals of the continuous problems. The minimizers of the discrete functionals provide converging approximations to the solution of the limit problem in question. Thus, we obtain approximation schemes that are nonconforming, but direct, and that can be treated by current algorithms for symmetric and positive definite functionals.The class of problems considered in this paper includes the Stokes problem in fluid dynamics, the loading problem of 2-D-isotropic elastostatics, and some boundary value problems of the Kirchhoff–Love theory of plates. Also discussed is an extension of the discretization method that seems suitable for treating more general boundary value problems of elastic plates, but whose convergence is conditional to a conjecture that remains to be proved. A relevant application to the so-called Babusˇka paradox is presented. [Copyright &y& Elsevier]

This paper is concerned with developing a medius error analysis for several nonconforming virtual element methods (VEMs) for the Poisson equation and the biharmonic equation in two dimensions, with the family of polygonal meshes satisfying a very general geometric assumption given in Brezzi et al. (2009) and Chen and Huang (2018). After some technical derivation, the inverse inequalities and norm equivalence are derived for some conforming VEMs. With the help of these results and following some ideas in Gudi (2010), we obtain medius error estimates for the nonconforming VEMs under discussion, which are optimal up to the regularity of the weak solution. Such estimates also imply that the nonconforming VEMs are convergent even if the exact solution only belongs to the admissible space while the right-hand side of the related equation has some additional regularity. [ABSTRACT FROM AUTHOR]

To construct C 1 -continuous basis functions for the numerical solutions of two dimensional biharmonic equations on non-convex domains with clamped and/or simply supported boundary conditions, we use B-spline basis functions instead of conventional Hermite finite element basis functions. The C 1 -continuous B-spline approximation functions constructed on the master patch are moved onto a physical domain by a geometric patch mapping. However it is difficult to cover a non-convex polygonal domain by one smooth patch mapping. Hence we decompose a non-convex domain into several overlapping rectangular subdomains and use the Schwartz alternating method to assemble local solutions for the global solution. Furthermore, in order to handle the corner singularities arising in non-convex domains, we introduce the implicitly enriched Galerkin method, which is similar to the explicit enrichment techniques used in XFEM, G-FEM, PUFEM, that are successful in handling the singularity problems arising in second-order differential problems. • Higher order B-spline functions are used for basis functions of 4th order problems. • To deal with corner and/or crack singularities, we introduce Implicitly Enriched GM. • To lessen computational complexity, DDM is combined with Implicitly Enriched GM. • To reduce computational cost, we introduce a Supplemental Subdomain Method. [ABSTRACT FROM AUTHOR]

In this paper, we study a problem of finding the solution for the nonlinear biharmonic equation Δ 2 u = f (x , t , u (x , t)) from the final data. By using a simple example, the ill-posedness of the present problem with random noise is demonstrated. The Fourier method is conducted in order to establish an estimator for the mild solution (called regularized solution) and the convergence results in some different cases are proposed. Finally, numerical experiments are presented for showing that this regularization method is flexible and stable. [ABSTRACT FROM AUTHOR]

A new weak Galerkin (WG) finite element method is designed featuring in using the first order polynomial to approximate solution of biharmonic equation. The proposed P 1 WG method achieves O (h) convergence in energy norm and O (h 2) in L 2 norm in solving the biharmonic equation. This is not possible for the traditional finite element method as the minimum polynomial degree is 2 in order to approximate the biharmonic equation. Numerical tests on various polygonal meshes verify the theory. [ABSTRACT FROM AUTHOR]