Abstract
Spaces of discrete differential forms can be applied to numerically solve the partial differential equations that govern phenomena such as electromagnetics and fluid mechanics. Robustness of the resulting numerical methods is complemented by pointwise satisfaction of conservation laws (e.g., mass conservation) in the discrete setting. Here we present the construction of isogeometric discrete differential forms, i.e., differential form spaces built using smooth splines. We first present an algorithm for computing Bézier extraction operators for univariate spline differential forms that allow local degree elevation. Then, using tensor-products of the univariate splines, a complex of discrete differential forms is built on meshes that contain polar singularities, i.e., edges that are singularly mapped onto points. We prove that the spline complexes share the same cohomological structure as the de Rham complex. Several examples are presented to demonstrate the applicability of the proposed methodology. In particular, the splines spaces derived are used to simulate generalized Stokes flow on arbitrarily curved smooth surfaces and to numerically demonstrate (a) optimal approximation and inf–sup stability of the spline spaces; (b) pointwise incompressible flows; and (c) flows on deforming surfaces.
Highlights
Partial differential equations (PDEs) describing physical phenomena are built on a rich differential and geometric foundation of conservation laws, topological constraints, symmetries and invariants
The formulation of such numerical methods is our focus in this article, with a special attention towards high-order accurate discretizations of PDEs defined on surfaces in R3
The numerical tests presented here show that the polar spline spaces demonstrate optimal approximation; similar results were obtained for the configurations not shown here
Summary
Partial differential equations (PDEs) describing physical phenomena are built on a rich differential and geometric foundation of conservation laws, topological constraints, symmetries and invariants. The extensions of finite element exterior calculus to isogeometric analysis have come via the development of isogeometric discrete differential form spaces, i.e., discrete differential form spaces built using smooth splines. The motivation for this article is construction of isogeometric discrete differential forms for numerical approximation of (scalar and vector) solutions to PDEs. We focus on discrete differential form spaces built using two particular classes of nonstandard spline spaces — univariate multi-degree splines and bivariate polar splines. We develop novel isogeometric discrete differential forms that, in particular, offer a high-order alternative to the above methods for simulation of flows on smooth surfaces without any recourse to Lagrange multipliers or penalties for enforcing tangentiality of the flow. We demonstrate the high-order accuracy, stability and applicability of the discrete differential form spaces by simulating, in particular, generalized Stokes flow on fixed and deforming smooth surfaces (Section 6). See an example of such pointwise incompressible tangential flow on the right where streamfunction contours and tangential velocities are displayed; see Section 6.3 for details
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