Abstract
The Virtual Element Method (VEM) is an evolution of the mimetic finite difference method which overcomes many limitations affecting classic Finite Element Methods (FEM). VEM for 2D problems allows for exploiting meshes consisting of any polygonal elements. No limitations on their internal angles are needed. Hanging nodes are easily treated. Notably, VEM is well apt to meshâadaptive algorithms. In this paper we detail an implementation of meshâadaptive VEM for potential problems. We suggest a fresh, promising approach. We show on suitable test problems that a gain in efficiency can be obtained, respect to uniform, fine discretizations.
Highlights
In the last two decades, the numerical treatment of partial di erential equations (PDEs) has been focused on treating meshes with arbitrarily-shaped polygonal/polyhedral elements
It is worth mentioning that a peculiar feature of Virtual Element method (VEM) is designing approximation spaces characterized by high continuity properties; For details see cf. [17] and the works on high-order partial di erential equations as the biharmonic equations mentioned above
We use an adaptive algorithm for elliptic problems consisting of the classic steps: solve, estimate, mark, refine [36]
Summary
In the last two decades, the numerical treatment of partial di erential equations (PDEs) has been focused on treating meshes with arbitrarily-shaped polygonal/polyhedral (polytopal, for short) elements. We use an adaptive algorithm for elliptic problems consisting of the classic steps: solve, estimate, mark, refine [36]. In this context, given a polygonal subdivision of the problem domain, one solves the VEM problem, estimates the error using our a posteriori error bound, marks a subset of elements for refinement, and refines marked elements. The consistency matrix can be computed, while the stability matrix is not computable The latter is approximated by introducing a local symmetric positive definite, element–wise bilinear form SE (·, ·).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: International Journal of Analytical, Experimental and Finite Element Analysis (IJAEFEA)
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
