Problem-oriented Virtual Element Methods

In this project, we deal with the approximation of solutions to certain differential problems, modelling physical phenomena ranging from wave propagation, electrostatic, and crack propagation. The underlying method we will employ is called the virtual element method, which is a generalization of the more traditional finite element method to polygonal and polyhedral meshes. The main goal of this project is to introduce novel techniques, in order to tackle the above mentioned problem in an efficient way. For instance, if we know that the exact solution behaves like a wave, we consider finite dimensional approximation spaces, which contain oscillating functions. Instead, if the exact solution has a singular behaviour, we consider finite dimensional spaces, which contain singular functions. Additionally, we investigate so-called adaptive methods for the approximation of singular solutions, via flux reconstruction. Roughly speaking, an adaptive method is a method that try to catch the behaviour of the exact solution using as few parameters as possible. The flux reconstruction technique is novel in the setting of the virtu al element method, and allows for an improvement of other existing techniques in literature.

In this project, we addressed the challenge of approximating solutions to a range of partial differential problems commonly encountered in the modeling of physical phenomena, including elasticity, heat conduction, electromagnetics, and magnetohydrodynamics. The core approach was the Virtual Element Method (VEM), a generalization of the Finite Element Method (FEM) that extends its applicability to polytopic meshes. This flexibility allows the VEM to model complex geometries more effectively and achieve higher accuracy, especially in cases where the traditional FEM struggles with nonstandard or irregular domains. The main objective was to develop and implement novel techniques to enhance the efficiency and accuracy of the VEM when applied to the aforementioned physical problems. Specifically, we focused on developing refined approximation spaces tailored to the characteristics of the exact solution. For problems where the solution exhibits singularities, such as those arising in elasticity, we employed finite-dimensional VE spaces incorporating singular functions to capture the solution's singularities with greater precision, thereby improving the overall accuracy of the approximation. For parabolic problems, such as heat conduction problems, we introduced VE spaces designed to capture both space and time evolution in a monolithic way, improving accuracy and stability across the entire solution domain by better handling the coupling between spatial and temporal variables. Another key aspect of the project was the investigation of adaptive methods for approximating singular solutions, particularly through the application of flux reconstruction techniques. Adaptive methods are designed to adjust the computational effort based on the local behavior of the solution, minimizing the number of parameters needed to achieve an accurate approximation. This is particularly useful in capturing sharp gradients or singularities with minimal computational cost. The flux reconstruction technique applied within the framework of the VEM was another central innovation in our work. Through these developments, we improved the efficiency, accuracy, and flexibility of the VEM for solving a broad range of differential problems. This has resulted in a more powerful tool for researchers and engineers working with complex physical systems, enabling high-fidelity simulations of real-world phenomena across engineering and physics.