Biharmonic Equation Research Articles

Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous Pk polynomials are employed. To connect discontinuous functions, the inter-element traces, {uh} and {∇uh}, 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 Pk+4 polynomial from four Pk 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 Pk solution has an O(hk+5) convergence in L2-norm, and an O(hk+3) convergence in H2-norm. A local post-process is defined, which lifts the Pk solution to a Pk+4 quasi-optimal solution. Numerical tests are provided, confirming the theory.

A Study of Biharmonic Equation Involving Nonlocal Terms and Critical Sobolev Exponent

AbstractIn this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of the equation $$\begin{aligned} \Delta ^{2} u= \Big (|x|^{-\theta }*|u|^{p_{\theta }}\Big )|u|^{p_{\theta }-2}u +\alpha \Big (|x|^{-\gamma }*|u|^{p}\Big )|u|^{p-2}u \quad \text{ in } {{\mathbb {R}}}^{N}, \end{aligned}$$ Δ 2 u = ( | x | - θ ∗ | u | p θ ) | u | p θ - 2 u + α ( | x | - γ ∗ | u | p ) | u | p - 2 u in R N , where $$0<p\le p_{\gamma }^{*}$$ 0 < p ≤ p γ ∗ , $$\alpha >0$$ α > 0 , $$\theta , \gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$p_{\theta }=\frac{2(N-\theta )}{N-4}$$ p θ = 2 ( N - θ ) N - 4 and $$N\ge 5$$ N ≥ 5 . Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.

Existence of Solutions for Biharmonic Equations on Conical Singular Manifolds

Existence of Solutions for Biharmonic Equations on Conical Singular Manifolds

Diagonalization based Parallel-in-Time method for a class of fourth order time dependent PDEs

In this paper, we design, analyze and implement efficient time parallel methods for a class of fourth order time-dependent partial differential equations (PDEs), namely the biharmonic heat equation, the linearized Cahn–Hilliard (CH) equation and the nonlinear CH equation. We use a diagonalization technique on all-at-once system to develop efficient iterative time parallel methods for investigating the solution behaviour of the said equations. We present the convergence analysis of Parallel-in-Time (PinT) algorithms. We verify our findings by presenting numerical results.

The local boundary knots method for solution of Stokes and the biharmonic equation

The paper focuses on the derivation of a local variant of the boundary knot method (BKM) for the cases of Stokes flow and the biharmonic equation. Compared to the global variant, the local one leads to a sparse result matrix of the system of equations and thus makes the solution of especially large-scale problems more efficient. It is also important to keep the conditionality of the interpolation matrix within reasonable bounds. For the localization, a combination of BKM and finite collocation method was used. The results of the local variant were compared on several examples and the dependence of the solution on the density of the point network and the dimensions of the stencil used was also tested in the paper.

LOCAL MODEL OF THE THERMOELASTIC STATE OF A POROUS MATERIAL

In the paper, a local stationary model of the thermoelastic state of a porous material is constructed. The model is based on uncoupled boundary value problems for stationary equations of heat conduction and thermoelasticity for a space with two spherical cavities. The temperature field is determined by the constant temperature on the surfaces of the cavities, which are considered free of forces. The problem was solved by the generalized Fourier method (GFM), for which the paper presents its further development for a certain class of thermoelasticity problems. For this purpose, systems of equally directed spherical coordinates were introduced, the origins of which are connected with the centers of the cavities. In the work, a new set of axisymmetric basic solutions of the Lamé equation for a sphere is constructed and addition theorems are proved for it and for solutions of the vector biharmonic equation in the given coordinate systems. The GFM formalism made it possible to reduce the boundary value problems to algebraic solvable systems with Fredholm operators in the space l2 under the condition that the spherical surfaces do not intersect. The reduction method was used for the numerical solution of the systems. Graphs of stresses on the surface of one of the cavities and stresses on the axis of symmetry of the problem between the cavities at different relative sizes of the cavities and at different temperatures of their heating were obtained. The obtained results agree with those known for one cavity. The convergence of the reduction method is verified numerically.

On the solution of a complicated biharmonic equation in a hydroelasticity problem

On the solution of a complicated biharmonic equation in a hydroelasticity problem

Conforming and nonconforming virtual element methods for fourth order nonlocal reaction diffusion equation

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer’s fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples.

A note on a critical bi-harmonic equation with logarithmic perturbation

In this short note, a critical bi-harmonic elliptic equation with logarithmic perturbation is considered. The same problem has been investigated recently by Li et al. (2023), where existence and nonexistence of weak solutions were obtained when the parameters and the domain satisfy some appropriate conditions. Part of their existence condition is weakened in this note. Moreover, the types of the weak solutions can be specified with the help of Brézis–Lieb’s lemma and Ekeland’s variational principle.

Existence of Solutions on the Critical Hyperbola for a Pure Lane–Emden System with Neumann Boundary Conditions

Abstract We study the following Lane–Emden system: $$\begin{align*} & -\Delta u=|v|^{q-1}v \quad \ \textrm{in}\ \Omega, \qquad -\Delta v=|u|^{p-1}u \quad \ \textrm{in}\ \Omega, \qquad u_{\nu}=v_{\nu}=0 \quad \ \textrm{on}\ \partial \Omega, \end{align*}$$with $\Omega $ a bounded regular domain of ${\mathbb{R}}^{N}$, $N \ge 4$, and exponents $p, q$ belonging to the so-called critical hyperbola $1/(p+1)+1/(q+1)=(N-2)/N$. We show that, under suitable conditions on $p, q$, least-energy (sign-changing) solutions exist, and they are classical. In the proof we exploit a dual variational formulation, which allows to deal with the strong indefinite character of the problem. We establish a compactness condition which is based on a new Cherrier-type inequality. We then prove such condition by using as test functions the solutions to the system in the whole space and performing delicate asymptotic estimates. If $N \ge 5$, $p=1$, the system above reduces to a biharmonic equation, for which we also prove existence of least-energy solutions. Finally, we prove some partial symmetry and symmetry-breaking results in the case $\Omega $ is a ball or an annulus.

On the solution of a complicated biharmonic equation in a hydroelasticity problem

A hydroelastic problem of free vibrations of a thin plate that horizontally separates ideal incompressible liquids of different densities in a rigid cylindrical tank with an arbitrary cross-section has been considered in the linear formulation. To solve the corresponding complicated inhomogeneous biharmonic equation, the fundamental system of the solutions of biharmonic equation (FSS) and the eigenmodes of ideal liquid oscillations in a cylindrical cavity were used. The frequency equation was obtained for arbitrary fixation of the plate contour. On the example of a clamped plate, the frequency equation was simplified by decomposing the corresponding homogeneous biharmonic equation into two harmonic equations and using Green's formula for the Laplace operator. It was shown that in this case the frequency equation does not depend on the FSS and becomes greatly simplified because the FSS depends on the unknown frequency. The resulting equation has a single form for the cases of a right circular cylinder and a rectangular channel; in particular cases, it coincides with the previously obtained equations. Research of asymmetric vibration frequencies of a plate and a membrane, as well as axisymmetric vibration frequencies of a membrane in a circular cylinder, has been carried out. An approximation formula for high frequencies and approximate conditions for the stability of the plate and membrane vibrations were obtained.

Global solution and blow-up for a class of Δγ-biharmonic parabolic equations with logarithmic nonlinearity

In this paper, we study a class of [Formula: see text]-biharmonic parabolic equations with logarithmic nonlinearity. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique and the Sobolev inequality, we obtain the existence of the unique global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

An Invariance and Closed Form Analysis of the Nonlinear Biharmonic Beam Equation

In this paper, we study the one-parameter Lie groups of point transformations that leave invariant the biharmonic partial differential equation (PDE) uxxxx+2uxxyy+uyyyy=f(u) . To this end, we construct the Lie and Noether symmetry generators and present reductions of biharmonic PDE. When f is arbitrary function of u, we obtain the solution of biharmonic equation in terms of Green function. The equation is further analysed when f is exponential function and for general power law. Furthermore, we use Noether's theorem and the 'multiplier approach' to construct conservation laws of the PDE.

Generalized weak Galerkin finite element methods for biharmonic equations

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 H2 norm and a L2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equation.

Solutions to a (p(x),q(x))-biharmonic elliptic problem on a bounded domain

Using variational methods and critical point results, we prove the existence and multiplicity of weak solutions of a (p(x),q(x))-biharmonic elliptic equation along with a singular term under Navier boundary conditions.

σ-monogenic functions in commutative algebras

In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.

High-Accuracy Numerical Schemes for Solving Plane Boundary Problem for a Polyharmonic Equation and Their Application to Problems of Hydrodynamics

Boundary value problems are considered for harmonic, biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on the plane. The problems are reduced to solving linear integral equations on boundary contours, which are assumed to be smooth. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which integral equations are expressed. The algorithm uses the periodicity of functions defined on closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed degree. Applications to solving problems of hydrodynamics, filtration and other problems of theoretical physics are considered.

Построение С1-гладких кусочно-квадратичных функций при решении краевых задач уравнений 4-го порядка на треугольной сетке

In this paper, we present one approach to constructing continuously differentiable piecewise-quadratic functions on a triangular grid, based on smoothing a piecewise-linear function in the vicinity of edges and triangulation nodes. In the works [8], [7], the issues of approximation of the functional (1) in triangular grids and the convergence of the variational method for solving the boundary value problem of the equation (2) were studied. However, in the numerical solution there are difficulties associated with the construction of continuously differentiable piecewise polynomial functions on triangulations. In particular, their construction requires solving large systems of equations at each step of the variational method. When trying to get by with only continuous piecewise polynomial functions, we got a negative result (divergence of approximate solutions was found) [9]. In this paper, we circumvent the difficulties that arise — we indicate a method for constructing piecewise polynomial functions that have continuous partial derivatives. With the help of this class of functions, we have obtained formulas for approximating the functional (1) and tested the method on the example of a biharmonic equation.

Positive solutions for a class of biharmonic equations: Existence and uniqueness

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

Hilbert Complexes: Analysis, Applications, and Discretizations

In this workshop 70 (43 at MFO, 27 online) leading mathematicians from Europe, United States, China, and Australia met at the MFO to discuss and present new developments in the mathematical and numerical analysis including discretizations of Hilbert complexes related to systems of partial differential equations, in particular the well known de Rham complex and the complexes of elasticity and the biharmonic equations. The report at hand offers the extended abstracts of their talks.