Existence of Positive Solutions for a Pertubed Fourth-Order Equation

In this paper, a special type of fourth-order differential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the differential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

Sino–German Computational and Applied Mathematics

This short article serves as an epilog of the thirteen preceding papers in this special issue of CMAM. All contributions are authored by participants of the 7th Sino–German Workshop on Computational and Applied Mathematics at the Kiel University. The topics cover fourth-order problems, solvers and multilevel methods, a posteriori error control and adaptivity, and data science.

Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.