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This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper. [ABSTRACT FROM AUTHOR]

An abstract property (H) is the key to a complete a priori error analysis in the (discrete) energy norm for several nonstandard finite element methods in the recent work [Lowest-order equivalent nonstandard finite element methods for biharmonic plates, Carstensen and Nataraj, M2AN, 2022]. This paper investigates the impact of (H) to the a posteriori error analysis and establishes known and novel explicit residual-based a posteriori error estimates. The abstract framework applies to Morley, two versions of discontinuous Galerkin, C0 interior penalty, as well as weakly over-penalized symmetric interior penalty schemes for the biharmonic equation with a general source term in H−2 (Ω). [ABSTRACT FROM AUTHOR]

The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics and has wide applications. In this paper, for the biharmonic eigenvalue problem, based on the work of Gudi [Numer. Methods Partial Differ. Equ., 27 (2011), 315-328], we study the a posteriori error estimates of the approximate eigenpairs obtained by the Ciarlet-Raviart mixed finite element method. We prove the reliability and efficiency of the error estimator of the approximate eigenfunction and analyze the reliability of the error estimator of the approximate eigenvalues. We also implement the adaptive calculation and exhibit the numerical experiments which show that our method is efficient and can get an approximate solution with high accuracy. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete H² norm is established for the corresponding finite element solutions. Error estimates in the L² norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with the p (x) -biharmonic operator, where the exponent p (x) is a given function. We transform the problem into a system of two differential equations and prove the existence, uniqueness and regularity of the weak solution. Then we formulate the discrete problem associated with that system using the finite element method and prove the existence, uniqueness and stability of the discrete solution. We also investigate the order of convergence and prove some error estimates. Next, we apply the Lagrange basis to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software considering the one and two dimensional cases and we present examples to illustrate the theory. • Study of a nonlinear beam equation with the p (x) -biharmonic operator. • Proof of the existence, uniqueness and regularity of the weak solution. • Discrete formulation with the mixed finite element method. • Demonstration of the existence, uniqueness and regularity of the discrete solution. • Numerical simulations with Matlab software for the one and two dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A |$C^0$| finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze a stabilizer-free C0 weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are C0 continuous piecewise polynomials of degree k + 2 with k ≽ 0 and in terms of face unknowns which are discontinuous piecewise polynomials of degree k + 1. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the C1 conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete H2-like norm and the H1 norm for k ≽ 0 are established for the corresponding WG finite element solutions. Error estimates in the L2 norm are also derived with an optimal order of convergence for k > 0 and sub-optimal order of convergence for k = 0. Numerical experiments are shown to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we use the fading regularization method to solve a biharmonic inverse problem, represented by the Cauchy problem. Two formulations are studied and implemented numerically using a finite element method (FEM). We present numerical reconstructions of the missing data on the inaccessible part of the boundary from the knowledge of over‐prescribed noisy data for both smooth and piecewise smooth two‐dimensional geometries. Numerical examples validate the convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data. [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 C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence. [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]

In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C 0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators. [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]

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realize the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues. [ABSTRACT FROM AUTHOR]

This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H 2 and standard L 2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces. [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 [13] , we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ 2 u = f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of f Ψ s k ⁎ − u Δ 2 Ψ s k ⁎ for a cut-off function Ψ and a dual singular function s k ⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u , we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ 2 u = f , we have to use C 1 -continuous basis functions. For continuous differentiable basis functions, we use either C 1 -continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities. [ABSTRACT FROM AUTHOR]

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α ∈ (0 , 1) is considered, whose solutions exhibit a weak singularity at initial time t = 0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p = − Δ u , then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H 1 -norm stability result of the method is established, and then a sharp H 1 -norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularization is considered to obtain a stable approximate solution. Error estimate is established for the regularized solution for different regularization schemes. Numerical results that confirm the theoretical results are also presented. [ABSTRACT FROM AUTHOR]

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side F ∈ H−2(Ω) replaced by F ○ (JIM) and then are quasi-optimal in their respective discrete norms. The smoother JIM is defined for a piecewise smooth input function by a (generalized) Morley interpolation IM followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj, ESAIM: M2AN49 (2015) 977–990.] without data oscillations. This paper extends the work [Veeser and Zanotti, SIAM J. Numer. Anal.56 (2018) 1621–1642] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms. [ABSTRACT FROM AUTHOR]

An initial-boundary value problem of the form D t α u + Δ 2 u − c Δ u = f is considered, where D t α is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain Ω ⊂ ℝ d for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of D t α on a graded temporal mesh. The numerical method computes approximations u h n and p h n of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on ∥ u h n ∥ L 2 (Ω) and ∥ p h n ∥ L 2 (Ω) ) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on ∥ u (⋅ , t n) − u h n ∥ L 2 (Ω) and ∥ Δ u (⋅ , t n) − p h n ∥ L 2 (Ω) are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → 1−. [ABSTRACT FROM AUTHOR]

• Construction of transparent boundary conditions (TBCs) for exural wave cavity scattering in two dimensions, ensuring the satisfaction of the sommerfeld radiation conditions. • Deduction of a decomposed problem for the biharmonic plate wave equation using two auxiliary functions and establishment of the unique-ness of the solution for the decomposed problem. • Proposal of the linear _nite element method with interior penalty term (IP-FEM) and boundary penalty term (BP-FEM) to solve the coupled boundary helmholtz and modi_ed helmholtz equations, leading to stable numerical solutions. • Introduction of an analytical solution for exural wave scattering by a circular cavity with a clamped boundary, facilitating comparative analysis. Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem. [ABSTRACT FROM AUTHOR]

A decoupled finite element method for the triharmonic equation in two dimensions is developed in this paper. The triharmonic equation can be decoupled into two biharmonic equations and one symmetric tensor-valued Stokes equation. Two biharmonic equations are discretized by the Morley element. An H 1 (S) -conforming finite element in arbitrary dimension is constructed and applied for solving the tensor-valued Stokes equation. The convergence analysis is presented for the resulting discrete method. Numerical results are provided to verify the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green's function estimates for the continuous problem which has independent interest. [ABSTRACT FROM AUTHOR]

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with |$L^\infty $| coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM. [ABSTRACT FROM AUTHOR]

A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and will reduce programming complexity. This stabilizer freeWG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete H² norm for k ≥ 2 and in an L² norm for k > 2 are established for the corresponding weak Galerkin finite element solutions, where k is the degree of the polynomial in the approximation. Numerical results are provided to confirm the theories. [ABSTRACT FROM AUTHOR]

It is shown that the first biharmonic boundary value problem on a topologically trivial domain in 3D is equivalent to three (consecutively to solve) second-order problems. This decomposition result is based on a Helmholtz-like decomposition of an involved non-standard Sobolev space of tensor fields and a proper characterization of the operator div Div acting on this space. Similar results for biharmonic problems in 2D and their impact on the construction and analysis of finite element methods have been recently published in Krendl et al. [A decomposition result for biharmonic problems and the Hellan–Herrmann–Johnson method. Electron Trans Numer Anal. 2016;45:257–282]. The discussion of the kernel of div Div leads to (de Rham-like) closed and exact Hilbert complexes, the div Div -complex and its adjoint the Grad grad -complex, involving spaces of trace-free and symmetric tensor fields. For these tensor fields, we show Helmholtz type decompositions and, most importantly, new compact embedding results. Almost all our results hold and are formulated for general bounded strong Lipschitz domains of arbitrary topology. There is no reasonable doubt that our results extend to strong Lipschitz domains in R N . [ABSTRACT FROM AUTHOR]

The weak Galerkin form of the finite element method, requiring only C0 basis function, is applied to the biharmonic equation. The computational procedure is thoroughly considered. Local orthogonal bases on triangulations are constructed using various sets of interpolation points with the Gram-Schmidt or Levenberg-Marquardt methods. Comparison and high-precision computations are carried out, and convergence rates are provided up to degree 11 for L2, 10 for H1, and 9 for H2, suggesting that the algorithm is useful for a variety of computations. [ABSTRACT FROM AUTHOR]

We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor. [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]

We introduce an hp-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and hp-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an arbitrary number of faces for polynomial basis with degree p = 2, 3. The key feature of the proposed method is that it employs elemental polynomial bases of total degree Pp, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes. [ABSTRACT FROM AUTHOR]

It is well known that convergence rate of finite element approximation is suboptimal in the L2 norm for solving biharmonic equations when P2 or Q2 element is used. The goal of this paper is to derive a weak Galerkin (WG) P2 element with the L2 optimal convergence rate by assuming the exact solution sufficiently smooth. In addition, our new WG finite element method can be applied to general mesh such as hybrid mesh, polygonal mesh or mesh with hanging node. The numerical experiments have been conducted on different meshes including hybrid meshes with mixed of pentagon and rectangle and mixed of hexagon and triangle. [ABSTRACT FROM AUTHOR]

We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both · and L2 norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces {[Pk (T)]2, Pk (e), Pk+1(T)} for k = 0 and the numerical convergence rates confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H4 = ϑ(w2) = ϑ(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness. [ABSTRACT FROM AUTHOR]

Abstract According to the extended Melan's static theorem, theoretical and numerical aspects of the stress compensation method (SCM) are presented to perform shakedown analysis of elastic-plastic structures considering the effect of temperature on yield strength. Instead of constructing a mathematical programming formulation, this developed method consists of the two-level iterative scheme. The inner loop constructs the statically admissible self-equilibrating stress field, while the outer loop evaluates a sequence of decreasing load factors to approach to the shakedown limit multiplier. The yield strength considering temperature effect is updated based on the current temperature at each outer iteration, and the yield conditions are checked at all Gauss points. The numerical procedure is well incorporated into ABAQUS finite element code and used for calculating the shakedown limits of structures considering yield strengths as different functions of temperature under complex thermomechanical loading system. The method is validated by some plane stress and axisymmetric numerical examples with theoretical and numerical solutions, and subsequently applied to solve the practical shakedown problem of a pipe with oblique nozzle. The results demonstrate that the developed method is stable, accurate and efficient, and can effectively evaluate the shakedown limit of an elastic-plastic structure where the yield strength of material varies with temperature. Graphical abstract Image 1 Highlights • An effective method is developed for shakedown analysis of structures with temperature-dependent yield strength. • Geometric illustration of extended static shakedown theorem considering temperature-dependent yield strength is given. • The developed method just performs some iterative calculations where the global stiffness matrix is decomposed only once. • 2D and 3D numerical examples considering the linear and nonlinear yield stress functions are solved and analyzed. • Failure modes and mechanisms considering the effect of temperature on shakedown boundaries are revealed. [ABSTRACT FROM AUTHOR]

We consider a saddle-point formulation for a sixth-order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet–Raviart formulation for the biharmonic problem to formulate our saddle-point problem and the finite element method. The new formulation allows us to use the |$H^1$| -conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods. [ABSTRACT FROM AUTHOR]

An algorithm is presented to solve the biharmonic problem for clamped plate by the Ritz method, with the use of explicit formulas for splines of degree 5 on a triangular grid of nodes. A computational experiment is carried out to solve the problem on a square domain for various partition schemes. [ABSTRACT FROM AUTHOR]

• For the biharmonic eigenvalue problem, Wang et al. [Adv. Comput. Math. 45(2019)] first analyzed the mixed DG method (MDG method) under the condition that the eigenfunction u and Δ u. have the same regularity. In this paper, we establish an MDG discretization which is different with that in the existing work and we conduct our analysis under the condition that u and Δ u have different regularities. • The existing literature derived the error estimate of biharmonic eigenvalue by using the estimate of Laplacian eigenvalue problem. In this paper, we establish an identity (see Lemma 3.5) and use it to get the a priori error estimate of biharmonic eigenvalue. We prove that the convergence rate of approximate eigenvalues reaches 2p-2 in h and p − (2 s − 7) in when p ≥ 2 and the eigenfunction u ∈ H S (Ω) (s ≥ p + 1) which is consistent with the estimate in the existing work [Babuska and Osborn, Eigenvalue problems, 1991] for the classical mixed finite element method for biharmonic eigenvalue problem. • Using the identity (see Lemma 3.5) we also discuss the a posteriori error estimate for the biharmonic eigenvalue. The literature on the a posteriori error estimate for Ciarlet-Raviart mixed method is still quite a few. We give an a posteriori error estimator of the approximate eigenvalue and analyze its reliability. The biharmonic equation/eigenvalue problem is one of the fundamental model problems in mathematics and physics, and has wide applications. In this paper, we study an hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Based on the work of Gudi et al. [J Sci Comput, 37 (2008)], using piecewise polynomials of degree p ≥ 3 , we derive the a priori error estimates of the approximate eigenfunction in the broken H 1 norm and L 2 norm which are optimal in h and suboptimal in p. When p = 2 , the approximate eigenfunctions converge but with only suboptimal convergence rate. When p ≥ 2 and the eigenfunction u ∈ H s (Ω) (s ≥ p + 1), we prove that the convergence rate of approximate eigenvalues reaches 2 p − 2 in h and p − (2 s − 7) in p. We also discuss the a posterior error estimates of the approximate eigenvalues and implement the adaptive calculation. Numerical experiments show that the methods are easy to implement and can efficiently compute biharmonic eigenvalues. [ABSTRACT FROM AUTHOR]

Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfürth for the related biharmonic equation. This article generalises Verfürth's result to Kirchhoff plates with an anisotropic material, which requires some modifications. Special emphasis is laid on the reduced Hsieh–Clough–Tocher triangular finite element, the conforming element with the least possible number of unknowns. [ABSTRACT FROM AUTHOR]

In this paper, some local multilevel methods are presented to solve the linear algebraic systems resulting from the application of adaptive conforming Bogner--Fox--Schmit rectangular element and nonconforming Adini rectangular element approximations to the biharmonic problem. The abstract Schwarz framework is applied to verify the uniform convergence of the local multilevel methods featuring Jacobi and Gauss--Seidel smoothing only on local nodes associated with the local refinements. By the abstract framework, a convergence estimate may also be derived from the stability of the space splitting and its strengthened Cauchy--Schwarz inequality. We demonstrate the optimality of the proposed algorithms by extensive numerical experiments. [ABSTRACT FROM AUTHOR]

The weak Galerkin (WG) methods have been introduced in Mu et al. (2013, 2014), [14] for solving the biharmonic equation. The purpose of this paper is to develop an algorithm to implement the WG methods effectively. This can be achieved by eliminating local unknowns to obtain a global system with significant reduction of size. In fact, this reduced global system is equivalent to the Schur complements of the WG methods. The unknowns of the Schur complement of the WG method are those defined on the element boundaries. The equivalence of the WG method and its Schur complement is established. The numerical results demonstrate the effectiveness of this new implementation technique. [ABSTRACT FROM AUTHOR]

We consider a new three-field formulation of the biharmonic problem. The solution, the gradient and the Lagrange multiplier are the three unknowns in the formulation. Adding a stabilization term in the discrete setting we can use the standard Lagrange finite element to discretize the solution, whereas we use the Raviart-Thomas finite element to discretize the gradient. The Lagrange multipliers are constructed to achieve the optimal error estimate. Numerical results are presented to demonstrate the performance of our approach. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 199-217, 2017 [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness effect. The stress results are compared with those provided by a recent theory which reduces the 3D governing equations of elasticity to a differential equation system, which includes a biharmonic equation and a harmonic equation. They provide the solution of the corresponding in-plane and out-of-plane notch problem, respectively, and have to be concurrently satisfied. Comparing numerical results and theoretical stress distributions, a good agreement is found. [ABSTRACT FROM AUTHOR]

The Ciarlet-Raviart mixed finite element method is popular for the biharmonic equations. In this paper, we apply this method to the Helmholtz transmission eigenvalue problem which is quadratic and non-self-adjoint, give a mixed variational form and a mixed method using the Lagrange elements, and prove the a priori error estimates of the discrete eigenvalues on the shape-regular meshes. We also give a spectral-mixed method. Theoretical analysis and numerical experiments show that these two methods are simple and easy to implement, and can efficiently compute real and complex transmission eigenvalues. [ABSTRACT FROM AUTHOR]

This article proposes and analyzes a C0-weak Galerkin (WG) finite element method for solving the biharmonic equation in two-dimensional and three-dimensional. The new WG method uses continuous piecewise-polynomial approximations of degree [ABSTRACT FROM AUTHOR]

Nonrigid or deformable 3D objects are common in many application domains. Retrieval of such objects in large databases based on shape similarity is still a challenging problem. In this paper, we take advantages of functional operators as characterizations of shape deformation, and further propose a framework to design novel shape signatures for encoding nonrigid geometries. Our approach constructs a context-aware integral kernel operator on a manifold, then applies modal analysis to map this operator into a low-frequency functional representation, called fast functional transform, and finally computes its spectrum as the shape signature. In a nutshell, our method is fast, isometry-invariant, discriminative, smooth and numerically stable with respect to multiple types of perturbations. Experimental results demonstrate that our new shape signature for nonrigid objects can outperform all methods participating in the nonrigid track of the SHREC'11 contest. It is also the second best performing method in the real human model track of SHREC'14. [ABSTRACT FROM AUTHOR]