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The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

In this paper, we study the phenomenon of increasing stability in the inverse boundary value problems for the biharmonic equation. The inverse problem is to determine the potential from DtN map. It is a kind of nonlinear inverse problem. By considering a linearized form, we obtain an increasing Lipschitz‐like stability when k$$ k $$ is large. Furthermore, we extend the discussion to the linearized inverse biharmonic potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. [ABSTRACT FROM AUTHOR]

In this article, high‐frequency stability estimates are explored for the determination of the zeroth‐order perturbation of the biharmonic operator with constant attenuation from the linearized partial Dirichlet‐to‐Neumann map when part of the boundary is inaccessible and flat. The results obtained suggest improvement of the stability with an appropriate choice of frequency. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the L 2 norm. A series of numerical experiments are conducted to validate the theoretical developments. [ABSTRACT FROM AUTHOR]

We investigate some nonlinear elliptic problems of the form \Delta^2v + \sigma(x) v= h(x,v)\quadin \Omega,\quad v=\Delta v=0 \quadon \partial\Omega, \eqno({\rm P}) where $\Omega$ is a regular bounded domain in $\mathbb{R}^N$, $N\geq2$, $\sigma(x)$ a positive function in $L^{\infty}(\Omega)$, and the nonlinearity $h(x,t)$ is indefinite. We prove the existence of solutions to the problem (P) when the function $h(x,t)$ is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

The paper investigates the approximate properties of biharmonic Poisson integrals for the upper half-plane on the classes of Hölder functions in the uniform metric. The exact values for the upper bounds of the deviations of the functions of the Hölder class H1 from the biharmonic Poisson integrals defined in the upper half-plane were found. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic Choquard-type problem Δ 2 u - β Δ u = λ u + (I μ ∗ F (u)) f (u) , in R 4 , ∫ R 4 | u | 2 d x = c 2 > 0 , u ∈ H 2 (R 4) , where β ≥ 0 , λ ∈ R , I μ = 1 | x | μ with μ ∈ (0 , 4) , F(u) is the primitive function of f(u), and f is a continuous function with exponential critical growth. By using the mountain-pass argument, we prove the existence of radial ground-state solutions for the above problem. [ABSTRACT FROM AUTHOR]

We investigate some nonlinear elliptic problems of the form (P) ∆2 v + σ(x)v = h(x, v) in Ω, v = ∆v = 0 on ∂Ω, where Ω is a regular bounded domain in RN, N > 2, σ(x) a positive function in L∞(Ω), and the nonlinearity h(x, t) is indefinite. We prove the existence of solutions to the problem (P) when the function h(x, t) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical. [ABSTRACT FROM AUTHOR]

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen [SIAM J. Sci. Comput., 44 (2022), pp. A1205-A1226] recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polyg- onal domains. In this paper, we present the LARS algorithm (lightning-AAA rational Stokes) for computing two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply con- nected domains using lightning and AAA rational approximation [Y. Nakatsukasa, O. Sète, and L. N. Trefethen, SIAM J. Sci. Comput., 40 (2018), pp. A1494-A1522]. After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error. [ABSTRACT FROM AUTHOR]

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems. [ABSTRACT FROM AUTHOR]

We consider the biharmonic equation Δ²u-(a + b R R5 |Δu|²dx) Δu + V (x)u = f(u), where V (x) and f(u) are continuous functions. By using a perturbation approach and the symmetric mountain pass theorem, the existence and multiplicity of solutions for this equation are obtained, and the power-type case f(u) = |u|p-2u is extended to p (2, 10), where it was assumed p (4, 10) in many papers. [ABSTRACT FROM AUTHOR]

In this paper, we address the Robin inverse problem for the biharmonic equation in a 2D$$ 2D $$ simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵2u=f(u,▵u) on an annular domain Ω in RN with the Navier boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:R+×R−→R+ is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f(ξ,η) to have superlinear or sublinear growth on ξ,η as |(ξ,η)|→0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones.

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]

We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally C 4 -smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L 2 -norm as well as to equivalents of the H s -seminorms for 1 ≤ s ≤ 4. In the studied case of spline degree p = 9 , the numerical results indicate in case of the Greville points a convergence of order O (h p − 3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O (h p − 2) for all (semi)norms except for the equivalent of the H 4 -seminorm, where the order O (h p − 3) is anyway optimal. [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 order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost- C 1 , Analysis-Suitable G 1 and the Approximate C 1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C 1 and Analysis-Suitable G 1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost- C 1 and D-Patch provide relatively easy construction on complex geometries. The Almost- C 1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C 1 and Analysis-Suitable G 1 applicable to more complex geometries. [ABSTRACT FROM AUTHOR]