Abstract
We consider a cut isogeometric method, where the boundary of the domain is allowed to cut through the background mesh in an arbitrary fashion for a second order elliptic model problem. In order to stabilize the method on the cut boundary we remove basis functions which have small intersection with the computational domain. We determine criteria on the intersection which guarantee that the order of convergence in the energy norm is not affected by the removal. The higher order regularity of the B-spline basis functions leads to improved bounds compared to standard Lagrange elements.
Highlights
Background and earlier work CutFEM and CutIGA, are methods where the geometry of the domain is allowed to cut through the background mesh in an arbitrary fashion, which manufactures so called cut elements at the boundary
New contributions We investigate the basis function removal approach based on eliminating basis functions that has a small intersection with the domain in the context of isogeometric analysis, more precisely we employ B-spline spaces of order p with maximal regularity Cp−1
Basis function removal can be done in a rigorous way which guarantees optimal order of convergence and that the resulting linear system is not arbitrarily close to singular
Summary
Background and earlier work CutFEM and CutIGA, are methods where the geometry of the domain is allowed to cut through the background mesh in an arbitrary fashion, which manufactures so called cut elements at the boundary. New contributions We investigate the basis function removal approach based on eliminating basis functions that has a small intersection with the domain in the context of isogeometric analysis, more precisely we employ B-spline spaces of order p with maximal regularity Cp−1. To this end we need to make the meaning of small intersection precise and our. When symmetric Nitsche is used to enforce Dirichlet boundary conditions stabilization appears to be necessary to guarantee that a certain inverse estimate holds This bound is not improved by the higher regularity of the splines and will not be enforced in a satisfactory manner by basis function removal. Outline In “The model problem and method” section we introduce the model problem and the method, in Chapter 3 we derive properties of the bilinear form, define the interpolation operator, define the criteria for basis function removal, derive error bounds, and quantify δ in terms of h for various norms, and in “Numerical results” section we present some illustrating numerical examples
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