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- Title
Virtual element method for elliptic bulk‐surface PDEs in three space dimensions.
- Authors
Frittelli, Massimo; Madzvamuse, Anotida; Sgura, Ivonne
- Abstract
In this work we present a novel bulk‐surface virtual element method (BSVEM) for the numerical approximation of elliptic bulk‐surface partial differential equations in three space dimensions. The BSVEM is based on the discretization of the bulk domain into polyhedral elements with arbitrarily many faces. The polyhedral approximation of the bulk induces a polygonal approximation of the surface. We present a geometric error analysis of bulk‐surface polyhedral meshes independent of the numerical method. Then, we show that BSVEM has optimal second‐order convergence in space, provided the exact solution is H2+3/4$$ {H}^{2+3/4} $$ in the bulk and H2$$ {H}^2 $$ on the surface, where the additional 34$$ \frac{3}{4} $$ is due to the combined effect of surface curvature and polyhedral elements close to the boundary. We show that general polyhedra can be exploited to reduce the computational time of the matrix assembly. Two numerical examples on the unit sphere and on the Dupin ring cyclide confirm the convergence result.
- Subjects
GEOMETRIC analysis; POLYHEDRA; POLYHEDRAL functions
- Publication
Numerical Methods for Partial Differential Equations, 2023, Vol 39, Issue 6, p4221
- ISSN
0749-159X
- Publication type
Article
- DOI
10.1002/num.23040