Fourth-order Problem Research Articles

A C0 finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

Abstract 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.

Global bifurcation curve for fourth-order MEMS/NEMS models

Global solution curve and exact multiplicity of positive solutions for a class of semilinear systems are obtained. The results extend some known global bifurcation results of P. Korman (2002) by allowing the nonlinearity to possess a singularity. Applications to fourth-order problems arising from MEMS/NEMS models give new global bifurcation results.

Approximate solutions of the Fourth-Order Eigenvalue Problem

In this paper, the differential transformation (DTM) and the Adomian decomposition (ADM) methods are proposed for solving fourth order eigenvalue problem. This fourth order eigenvalue problem has nonstrongly regular boundary conditions. This the fourth order problem has been examined for p(t) = t, B = 0, a = 0,01 where p(t) ≠ 0 is a complex valued and a ≠ 0 The differential transformation and the Adomian decomposition methods are briefly described. An approximate solution is obtained by performing seven iterations with the Adomian decomposition method. The same number of iterations have been made in the differential transformation method. The approximation results obtained by both methods have been compared with each other. These data have been presented in table. The ADM and the DTM approximation solutions have been shown by plotting in Figure 1. Here, the approaches obtained by using the two methods are found to be in high agreement. Consequently, highly accurate approximate solutions of fourth order eigenvalue problem are obtained. Such good results also revealed that the Adomian decomposition and the differential transformation methods are fast, economical and motivating. The exact solution of the fourth order eigenvalue problem for nonstrongly regular can not be found in the literature. Therefore, this study will give an important idea to determine approximate solution behavior of this fourth order problem.

Existence of three weak solutions for fourth-order Leray–Lions problem with indefinite weights

In this paper, we establish the existence of at least three weak solutions for the fourth-order problem with indefinite weights involving the Leray–Lions operator with nonstandard growth conditions. We generalize the result obtained by Kefi et al. [On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions. Bull Math Sci. 2021;DOI:10.1142/S1664360721500089]. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [On the structure of the critical set of nondifferentiable functions with a weak compactness condition. Appl Anal. 2010;89:1–10].

Existence results for perturbed fourth-order Kirchhoff type elliptic problems with singular term

Under appropriate growth conditions on the nonlinearity, the existence of multiple solutions for a perturbednonlocal fourth-order Kirchhoff-type problem involving the Hardy term:$$\Delta_p ^2 u-\big[M(\int_{\Omega}|\nabla u|^{p}dx)\big]^{p-1}\Delta_{p}u-\mu\frac{|u|^{p-2}u}{|x|^{2p}}= \lambda f(x,u),$$is established. Our main tools are based on variational methods and some critical pointstheorems. We give some examples to illustrate the obtained results.

Robust Preconditioning and Error Estimates for Optimal Control of the Convection--Diffusion--Reaction Equation with Limited Observation in Isogeometric Analysis

In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a method of discretization. To efficiently realize the preconditioner, we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.

On a p(x)- Kirchhoff fourth order problem involving Leray–Lions type operators

On a p(x)- Kirchhoff fourth order problem involving Leray–Lions type operators

An efficient finite element method and error analysis for fourth order problems in a spherical domain

<p style='text-indent:20px;'>An efficient finite element method for the fourth order problems in a spherical domain will be solved in this paper. Initially, we derive the necessary pole conditions with the intention of overcoming the difficulty of singularity introduced by spherical coordinate transformation. Then the original problem is transformed into a series of equivalent one-dimensional problems by utilizing spherical harmonic functions expansion. Secondly, we introduce some appropriate weighted Sobolev spaces and derive weak form and corresponding discrete form for each one-dimensional fourth order problem based on these pole conditions. In addition, we illustrate the error estimate of the approximate solutions by employing Lax-milgram lemma and approximation property of the cubic Hermite interpolation operator. Eventually, we present the algorithm in detail and show its efficiency through some numerical examples.</p>

Compact Difference Method for Time-Fractional Neutral Delay Nonlinear Fourth-Order Equation

In this paper, we present a compact finite difference method for a class of fourth-order nonlinear neutral delay sub-diffusion equations in two-dimensional space. The fourth-order problem is first transformed into a second-order system by a reduced-order method. Next by using compact operator to approximate the second order space derivatives and L2-1σ formula to approximate the time fractional derivative, the difference scheme which is fourth order in space and second order in time is obtained. Then, the existence and uniqueness of solution, the convergence rate of and the stability of the scheme are proved. Finally, numerical results are given to verify the accuracy and validity of the scheme.

Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities

In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.

Existence of multiple solutions for a fourth-order problem with variable exponent

<p style='text-indent:20px;'>We provide a new multiplicity result for a weighted <inline-formula><tex-math id="M1">\begin{document}$ p(x) $\end{document}</tex-math></inline-formula>-biharmonic problem on a bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula> with Navier conditions on <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>. Our approach, of variational nature, requires a suitable oscillating behavior of the nonlinearity and the associated weight to be compactly supported in <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>.</p>

An efficient finite-element method and error analysis for the fourth-order elliptic equation in a circular domain

In this paper, we propose an efficient finite-element method based on dimension reduction scheme to solve the fourth-order elliptic equation in a circular domain. First of all, by using polar coordinate transformation and Fourier basis function expansion, the necessary pole conditions are derived and the original problem is decomposed into a series of equivalent one-dimensional fourth-order problems. Based on the pole conditions, the appropriate weighted Sobolev spaces are introduced and the error estimate of finite-element solution is proved for each one-dimensional fourth-order problem. Then we construct the cubic Hermite interpolation basis functions and derive the matrix form corresponding to the discrete scheme. Furthermore, the sparsity of mass matrix and stiffness matrix are illustrated. Finally, we provide some numerical experiments, and the numerical results show that our method is very effective.

A family of C1 quadrilateral finite elements

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 }, 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.

Hybrid high-order method for singularly perturbed fourth-order problems on curved domains

We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilization operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.

On solution of one linear problem with initial and boundary conditions

We solve one boundary problem of fourth order with initial conditions, that appears, for example, when one solves the problem about lateral oscillations of elastic-viscous-relaxating rod of variable profile with variable momentum of inertia with freely supported ends.

On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray–Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [Appl. Anal. 89 (2010) 1–10].

Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems

In this paper, we consider positive supersolutions of the semilinear fourth-order problem { ( − Δ ) 2 u = ρ ( x ) f ( u ) in Ω , − Δ u > 0 in Ω , where Ω is a domain in R N (bounded or not), f : D f = [ 0 , a f ) → [ 0 , ∞ ) ( 0 < a f ⩽ + ∞ ) is a non-decreasing continuous function with f ( u ) > 0 for u > 0 and ρ : Ω → R is a positive function. Using a maximum principle-based argument, we give explicit estimates on positive supersolutions that can easily be applied to obtain Liouville-type results for positive supersolutions either in exterior domains, or in unbounded domains Ω with the property that sup x ∈ Ω ⁡ dist ( x , ∂ Ω ) = ∞ . In particular, we consider the above problem with f ( u ) = u p ( p > 0 ) and with different weights ρ ( x ) = | x | a , e a x 1 or x 1 m ( m is an even integer). Also, when f is convex and ρ : Ω → ( 0 , ∞ ) is smooth with Δ ( ρ ) > 0 , then under an extra condition between f and ρ we show that every positive supersolution u of this problem with u = 0 on ∂Ω (Ω bounded) satisfies the inequality − Δ u ≥ 2 ρ ( x ) F ( u ) for all x ∈ Ω , where F ( t ) : = ∫ 0 t ( f ( s ) − f ( 0 ) ) d s .

A [formula omitted] weak Galerkin method for linear Cahn–Hilliard–Cook equation with random initial condition

This paper introduces a C0 weak Galerkin finite element method for a linear Cahn–Hilliard–Cook equation. The highlights of the proposed method are that the complexity of constructing the C1 finite element space for fourth order problem is avoided and the number of degree of freedom is apparently reduced compared to the fully discontinuous weak Galerkin finite element method. With the redefined discrete weak Laplace operator and the classical C0 Lagrange elements, the L2 optimal error estimates in spatial variable are obtained. In time, the classical Euler scheme is then used to do the numerical simulation. Finally, numerical experiments are presented to demonstrate the efficiency of the proposed numerical method.

Removal of spurious outlier frequencies and modes from isogeometric discretizations of second- and fourth-order problems in one, two, and three dimensions

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, a few spurious modes appear that possess inaccurate frequencies, denoted as “outliers”. The outlier frequencies and corresponding modes are at the root of several efficiency and robustness issues in isogeometric analysis. One example is explicit dynamics where outlier frequencies unnecessarily reduce the critical time step. Another example is wave propagation where the inaccurate outlier modes may participate in the solution. In this paper, we first investigate the spurious outlier frequencies and corresponding modes of isogeometric discretizations of second- and fourth-order model problems and provide a complete characterization. We then devise a new approach that removes all outliers modes without negatively affecting the accuracy of the discretizations. Our approach is variationally consistent and works for a range of common boundary conditions on tensor product domains. We finally demonstrate that our approach allows a much larger critical time step, irrespective of polynomial degree, providing a pathway towards efficient higher-order explicit dynamics.

Morley finite element method for the von Kármán obstacle problem

This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.