Towards Virtual Element Methods for Moving Domain Problems
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Moving domain problems arise in many applications and present significant challenges for numerical discretizations, particularly when the computational mesh evolves over time. In this talk, we move towards the development of virtual element methods for elliptic problems posed on moving domains, where the computational grid is obtained by intersecting the boundary of the evolving domain with a fixed background Cartesian mesh. This construction naturally leads to time-dependent polygonal meshes whose geometry may change at each time step. As a first step towards the full moving-domain framework, we consider and analyse a model elliptic problem posed on a fixed domain discretized by an unfitted polygonal mesh generated through such an intersection procedure. The resulting meshes consist of general polygonal elements and may include elements exhibiting strong anisotropy. The unfitted nature of the discretization, combined with possible anisotropy of the elements, introduces nonstandard features that require special care in the theoretical analysis. We present a virtual element approximation tailored to this setting and investigate its stability and convergence properties under suitable assumptions on the mesh geometry. The analysis highlights the robustness of the virtual element framework when applied to polygonal and anisotropic meshes arising from unfitted discretizations. Numerical experiments are finally presented to illustrate the practical performance of the method and to confirm the theoretical results.
