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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.

An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ 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 ${D}_{t}^{\alpha }$ 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 $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$ ) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ 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−.

A new $$C^0$$ nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space $$P_3$$ and some bubble functions. Different from non- $$C^0$$ nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution $$u\in H_0^2(\Omega )\cap H^4(\Omega )$$ , the scheme is of linear order convergence by the measurement of discrete norm $$\Vert \cdot \Vert _h$$ . Some numerical results are included to further illustrate the convergence analysis.

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C 0 C^0 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.

Aim . To analyse the possibilities of various computational methods (primarily numerical) in terms of investigating the stress-strain state of complex multi-dimensional vertical loadbearing systems. Methods . The fundamental essence of the finite-difference method (grids) and the finite element method is revealed, their advantages and disadvantages are described in terms of solving the plane problem of elasticity theory, in particular, in calculating the stress-strain state of panelframe structures united in a single vertical multi-connected system. Results . The obtained results can be used to optimize the methodology of theoretical stress-strain state analysis of complex multiconnected systems, taking into account available computer equipment and licensed software packages for automated calculation of building structures. Conclusion . The conducted analysis shows that, provided there is a sufficiently powerful computer, the finite element method is the most versatile and effective method. This method was the basis of a large number of software packages permitting analysis of the stress-strain state of any designs characterized by the complexity of form, topology, load, etc.

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H 2 -conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases.

A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete $$H^2$$ norm are established for the corresponding finite element solutions. Error estimates in the $$L^2$$ 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. The numerical results are presented to confirm the theory of convergence.

In this article, a family of H2-nonconforming finite elements on tetrahedral grids is constructed for solving the biharmonic equation in 3D. In the family, the Pl polynomial space is enriched by some high order polynomials for all l ≥ 3 and the corresponding finite element solution converges at the order l − 1 in H2 norm. Moreover, the result is improved for two low order cases by using P6 and P7 polynomials to enrich P4 and P5 polynomial spaces, respectively. The error estimate is proved. The numerical results are provided to confirm the theoretical findings.

This paper proposes an efficient solution procedure for the deformation of a viscoelastic film pressed by another elastic film in cases wherein the films are on partially-opened substrates. The problem is formulated as a quasi-static stress state, where the stress equilibrium is satisfied at each time instance. The instantaneous stress equilibrium is expressed by biharmonic and harmonic functions for the stress and displacement of those general solutions that can be obtained in the Fourier domain. By applying the boundary conditions mathematically everywhere but the surface, which is in direct contact with the partially-opened substrate, the total problem could be discretized and solved using the Fourier collocation method. In this manner, the total number of unknowns can be greatly reduced. The validity and effectiveness of the proposed method was confirmed by comparing the results with those obtained by a finite element method (FEM) solver. The viscoelastic deformation obtained by the proposed method was almost in complete agreement with the result obtained by the FEM. The elapsed computational time required by the proposed method was approximately 1/100 of that required by the FEM.

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.

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 reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete $H^2$ for $k\ge 2$ and in $L^2$ norm for $k>2$ are established for the corresponding weak Galerkin finite element solutions. Numerical results are provided to confirm the theories., arXiv admin note: text overlap with arXiv:1309.5560, arXiv:1510.06001 by other authors

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson’s problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.

We present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C1 quadrilateral is based upon the construction of multi-patch C1 isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J. 72(692), 701–709 1968). Just as for the Argyris triangle, we additionally impose C2 continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 2019), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005), for polynomial degrees p = 3 and p = 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in $L^{\infty }$ L ∞ , L2, H1 and H2 for the piecewise-polynomial macro-element constructions of degree p ∈{3,4} and polynomial elements of degree p ≥ 5. Since the elements locally reproduce polynomials of total degree p, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des. 69, 55–75 30) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput. 22(1–3), 83–118 2005). In addition, we describe the construction of a simple, local basis and give for p ∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed C1 quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for p = 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.

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.

In this paper, a three-field formulation for the biharmonic equation is considered that consists of the potential u , the flux σ and a Lagrange multiplier. The use of stabilization techniques circumvents the discrete inf–sup condition and thus enables the application of arbitrary conforming finite element spaces for the three fields. A priori error estimates as well as reliable and efficient a posteriori error estimates are derived. The proof of the latter is based on an implicit, never to be computed, H 2 -reconstruction of the discrete potential. Several numerical experiments underline the theoretical results and show the existence of discretization spaces for which the a posteriori error estimate is p -robust with an efficiency index tending towards the optimal value of one. The numerical experiments indicate that the proposed method is applicable even if the theoretically required minimal regularity of the Lagrange multiplier for the a priori error estimate is not fulfilled. In particular, h - and h p -adaptivity recover optimal algebraic and exponential converge rates, respectively.

The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic $$H^3$$ -nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.

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.

We approximate the spectra of a class of 2 n -order differential operators using isogeometric analysis in mixed formulations. This class includes a wide range of differential operators such as those arising in elliptic, biharmonic, Cahn–Hilliard, Swift–Hohenberg, and phase-field crystal equations. The spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations, which require auxiliary parameters. The mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying B-spline space. We improve this order to be 2 p + 2 by applying optimally-blended quadrature rules developed in Puzyrev et al. (2017), Caloet al. (0000) and this order is an optimum in the view of dispersion error. We also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations.

This paper develops a new recovery-based linear C0 finite element method for approximating the weak solution of a fourth-order singularly perturbed Monge-Ampere equation, which is known as the vanishing moment approximation of the Monge-Ampere equation. The proposed method uses a gradient recovery technique to define a discrete Laplacian for a given linear C0 finite element function (offline), the discrete Laplacian is then employed to discretize the biharmonic operator appeared in the equation. It is proved that the proposed C0 linear finite element method has a unique solution using a fixed point argument and the corresponding error estimates are derived in various norms. Numerical experiments are also provided to verify the theoretical error estimates and to demonstrate the efficiency of the proposed recovery-based linear C0 finite element method.

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 regularisation is considered to obtain a stable approximate solution. Error estimate is established for the regularised solution for different regularisation schemes. Numerical results that confirm the theoretical results are also presented., Comment: 26 pages, 6 figures, 5 tables

We consider the boundary value problem for an inhomogeneous biharmonic equation and extend the problem in a variational form. Also, we carry out the discretization of the considered problem by the finite element method. The continued problems, is presented in matrix form. An iterative process is formulated for an approximate solution to the original problem. The problem in matrix form is solved by method of fictitious components. Estimates of the convergence of the iterative process are given. Research of the problem being solved is reduced to a cyber-physical system. The estimates of the convergence of the absolute error do not depend on the grid steps. An algorithm is formulated for solving the investigated computer problem. To process information in the algorithm, a variation method is used. The steepest descent method is used to control computations in the iterative process. The software implementation of the algorithm uses certain elements of artificial intelligence.

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.

For the biharmonic equation or this singularly-perturbed biharmonic equation, lower order nonconforming finite elements are usually used. It is difficult to construct high order $$C^1$$ conforming, or nonconforming elements, especially in 3D. A family of any quadratic or higher order weak Galerkin finite elements is constructed on 2D polygonal grids and 3D polyhedral grids for solving the singularly-perturbed biharmonic equation. The optimal order of convergence, up to any order the smooth solution can have, is proved for this method, in a discrete $$H^2$$ norm. Under a full elliptic regularity $$H^4$$ assumption, the $$L^2$$ convergence achieves the optimal order as well, in 2D and 3D. Numerical tests are presented verifying the theory.

We propose an additive average Schwarz preconditioner with two adaptively enriched coarse space for the nonconforming Morley finite element method for fourth order biharmonic equation with highly varying and discontinuous coefficients. In this paper, we extend the work of [9, 10]: (additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems). Our analysis shows that the condition number of the preconditioned problem is bounded independent of the jump of the coefficient, and it depends only on the ratio of the coarse to the fine mesh.

An approach to solving the biharmonic equation of the plane problem of the theory of elasticity by the numerical-analytical method of boundary elements is developed. The reduction of the two-dimensional problem to the one-dimensional one was carried out by the KantorovichVlasov method. Systems of fundamental orthonormal functions and the Green function are constructed without any restrictions on the nature of the boundary conditions. A numerical example of solving a plane problem by the boundary element method for a rectangular plate is considered. The results are compared with the data of finite element analysis in the ANSYS program and those obtained by A.V. Aleksandrov.

This paper develops a numerically effective approach to control synthesis for repetitive transverse loads in cantilever beam using an iterative learning control. In particular, the optimal tracking problem for the transverse displacements in the beam are considered. The system dynamics are modelled by a partial differential equation with a biharmonic operator with suitable boundary conditions at the clamped and free ends of the beam. The key novelty is the regulation of the temporal dynamics of the system state by a D-type iterative learning control law based on the second derivative of the beam displacement measured at a priori chosen reference spatial locations. The main systems theory contributions of this paper are the analysis of the convergence properties of control law and the development of a numerical scheme for implementation based on a finite element method. A physically relevant numerical example is given to illustrate the application of the new design in the form of dynamic displacement control for a vibrating cantilever beam.

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in L∞. The associated equation, coined the ∞-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by Δ2∞u:=(Δu)3|D(Δu)|2=0. In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call ∞-Biharmonic functions. For fixed p we design a mixed finite element scheme for the pre-limiting equation, the p-Bilaplacian Δ2pu:=Δ(|Δu|p−2Δu)=0. We prove convergence of the numerical solution to the weak solution of Δ2pu=0 and show that we are able to pass to the limit p→∞. We perform various tests aimed at understanding the nature of solutions of Δ2∞u and in 1-d we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of -solutions.

A numerical technique is applied to solve a biharmonic equation problem of a steady-state seepage flow of water through a thin layer of soil, under an impermeable dam. Due to the existence of a point boundary singularity and particular boundary conditions in the problem of the present study, the biharmonic equation is selected to be the governing equation. The potential function of the problem is approximated by the leading terms of the local asymptotic solution expansion. These terms are also used to weight the governing biharmonic equation in the Galerkin sense. Implementing Green’s theorem twice, the discretized equations are reduced to boundary integrals. The system of linear equations is completed by weakly enforcing the Dirichlet boundary conditions by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients. The numerical results obtained with the method, indicate high accuracy and very fast convergence with the number of singular functions and the number of Lagrange multipliers. The volume of water per unit time q flowing through the soil in the horizontal direction, under the dam, which is directly proportional to the directional derivative of the potential function, is approximated by the derivatives of the leading terms of the local asymptotic solution expansion. Values of q and the corresponding discharge velocity, obtained with the method, compare favourably with the values given by FEM and soil Mechanics theory. The favourable numerical behaviour of this numerical technique, which in the bibliography is known as singular function boundary integral method (SFBIM), can be exploited in numerous ways. Combining the SFBIM with standard schemes, such as the FEM, appears to be an interesting extension of the method for future work, aiming at enforcing the strengths of the SFBIM in problems with boundary singularities.

This paper proves the saturation assumption for the nonconforming Morley finite element discretization of the biharmonic equation. This asserts that the error of the Morley approximation under uniform refinement is strictly reduced by a contraction factor smaller than one up to explicit higher-order data approximation terms. The refinement has at least to bisect any edge such as red refinement or 3-bisections on any triangle. This justifies a hierarchical error estimator for the Morley finite element method, which simply compares the discrete solutions of one mesh and its red-refinement. The related adaptive mesh-refining strategy performs optimally in numerical experiments. A remark for Crouzeix-Raviart nonconforming finite element error control is included.

We propose Perfectly Matched Layers (PMLs) for flexural waves in plate structures. The analytical model is based on transformation optics techniques applied to the biharmonic fourth-order partial differential equation describing flexural vibrations in Kirchhoff–Love plates. We show that perfect boundary conditions are not an optimal solution, since they depend on the incident waves. The full analytical form of PMLs and zero reflection conditions at the boundary between homogeneous and PML domains are given. The implementation in a Finite Element (FEM) code is described and an eigenfrequency analysis is given as a possible methodology to check the implementation itself. A measure of the performances of the PMLs is introduced and the effects of element discretization, boundary conditions, frequency, dimension of the PML, amount of transformation and dissipation are detailed. It is shown that the model gives excellent results also when the applied load approaches the PML domain.

Residual error estimation for conforming finite element discretisations of the isotropic Kirchhoff plate problem is covered by an estimator of Verfurth for the related biharmonic equation. This article generalises Verfurth'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.