Adaptive Isogeometric Methods for Boussinesq Problems

Computational Fluid Dynamics (CFD) is the numerical study of fluid flow, heat transfer, turbulence modelling and several conservation laws. The fluids can be liquids, gases and even plasmas. There is a vast number of differential equations describing the physical problems in specific situations, and many of them have a nonlinear structure. The main goals of CFD are constructing an efficient and stable numerical technique for discretizing the equations with respect to space and time, and then solving the (nonlinear) algebraic systems of equations arising from the discretization as quick and accurate as possible. This might not be straightforward because the procedure always depends on the specific situation. The simulations are often carried out over long time intervals, and that will make high accuracy in space and time desirable. Furthermore, nonlinearity and local instabilities can also slow down the computational speed. When we couple several equations together, the solution procedure becomes even more complex. The most common numerical procedures utilized in CFD are the Finite Difference Method (FDM), the Finite Volume Method (FVM) and the Finite Element Method (FEM). The latter one is most the general and widespread because we can apply it on arbitrary complex domains that are sufficiently smooth, and it can perform local refinement in those parts of the domain where the unknown solution lacks sufficient regularity. There are a lot of similar FEM-approaches, and they differ most with respect to the choice of basis functions. One such method is called Isogeometric Analysis (IGA). It has superior approximation properties compared with classical FEM, and its signature ability is creating an exact mesh of the domain’s geometry. The discretized equations can be solved quickly, and all these advantages make IGA well-suited for CFD applications. The main focus of the thesis is solving the hydrodynamic Boussinesq equations for buoyancy-driven flow numerically. The PDE system consists of the Navier-Stokes equation and Advection-Diffusion equation coupled together. In particular, our research emphasizes adaptive error estimation and local refinement using isogeometric discretization. Adaptive refinement originated in the late 1970s. It was designed for reducing approximation error by generating a new mesh repeatedly until it resembled the unknown solution’s physical structure. In classical FEM, the theory of a posteriori error estimation is complete and has been applied widely to large classes of differential equations. This method is far better than a priori error estimation because it allows us to analyse local parts of the solution effectively and determine the corresponding local error. In CFD, there are many well-known situations where adaptive refinement and error estimation are desirable. We need a suitable method for reducing the error quickly without too much computational effort at the same time. We consider qualitative analysis of efficient a posteriori error estimators for IGA. This topic is still in a development stage although the classical refinement theory is compatible with the isogeometric paradigm. Splines are in general not interpolatory like the shape functions from FEM. Since they have higher continuity and better approximation properties, there is a good reason to believe that isogeometric refinement yields very good results for smooth problems. We will investigate whether some of these classical error estimators can be adapted directly to IGA, and then test them on some major PDEs in CFD: the Stokes equation, the Navier-Stokes equations, the Advection-Diffusion equation, and the Boussinesq equations