Numerical integration in the virtual element method with the scaled boundary cubature scheme
Copyright policy )Numerical integration in the virtual element method with the scaled boundary cubature scheme
AbstractThe virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.
- University of Milano-Bicocca Italy
- Los Alamos National Laboratory United States
- Lawrence Berkeley National Laboratory United States
- University of California, Davis United States
Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, cubature scheme; curved domain; nonpolynomial function; scaled boundary parametrization; simple polytope; weak singularity;, Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs, nonpolynomial function, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, Numerical quadrature and cubature formulas, Numerical computation using splines, scaled boundary parametrization, Biharmonic, polyharmonic functions and equations, Poisson's equation in two dimensions, curved domain, weak singularity, cubature scheme, simple polytope
Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, cubature scheme; curved domain; nonpolynomial function; scaled boundary parametrization; simple polytope; weak singularity;, Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs, nonpolynomial function, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for boundary value problems involving PDEs, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, Numerical quadrature and cubature formulas, Numerical computation using splines, scaled boundary parametrization, Biharmonic, polyharmonic functions and equations, Poisson's equation in two dimensions, curved domain, weak singularity, cubature scheme, simple polytope
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