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Partial differential equations (PDE) on surfaces appear in a variety of applications, such as image processing, modeling of lubrication, fluid flows, diffusion, and transport of surfactants. In some applications, surfaces are drawn and modeled by using CAD software, giving a very precise patchwise parametric description of the surface. This thesis deals with the development of methods for finding numerical solutions to PDE posed on such parametrically described multipatch surfaces. The thesis consists of an introduction and five papers. In the first paper, we develop a general framework for the Laplace-Beltrami operator on a patchwise parametric surface. Each patch map induces a Riemannian metric, which we utilize to compute quantities in the simpler reference domain. We use the cut finite element method together with Nitsche’s method to enforce continuity over the interfaces between patches. In the second paper, we extend the framework to be able to handle geometries that consist of an arrangement of surfaces, i.e., more than two per interface. By using a Kirchhoff's condition this method avoids defining any co-normal to each surface and can deal with sharp edges. This approach is shown to be equivalent to standard Nitsche interface method for flat geometries. In the third paper, we developed a cut finite element method for elliptic problems with corner singularities. The main idea is to use an appropriate radial map that grades the finite element mesh towards the corner that counter-acts the solution's singularity. In the fourth paper, we present a new robust isogeometric method for surfaces described by CAD patches with gaps or overlaps. The main approach here is to cover all interfaces with a three-dimensional mesh and then use a hybrid variable in a Nitsche-type formulation to transfer data over the gaps. Using this hybridized approach leads to a convenient and easy to implement method with no restriction on the number of coupled patches per interface. In the fifth paper, we present a routine to the multipatch isogeometric framework for dealing with singular maps. To exemplify this, we consider a specific type of singular parametrization which essentially maps a square onto a triangle. One part of the boundary of the square will be transformed into a single point and the metric tensor becomes singular as we approach this boundary. In this work we propose a regularization procedure which is based on eigenvalue decomposition of the metric tensor. |
