Optimal isogeometric boundary element methods

The ultimate goal of any numerical scheme is to compute a discrete solution with error below a prescribed tolerance at the expense of, up to a multiplicative constant, the minimal computational cost. In computational PDEs, the convergence behavior of numerical schemes is, however, usually spoiled by singularities of the given data and/or the unknown solution. One remedy is to use adaptive strategies. Although adaptive strategies are successfully employed since the eighties, their mathematical understanding is still in its infancy and restricted to model problems and standard methods like the finite element method (FEM) or the boundary element method (BEM) with piecewise polynomials of fixed polynomial order. The central idea of isogeometric analysis (IGA) is to use the same ansatz functions for the discretization of the PDE at hand as are used for the representation of the problem geometry, which in particular avoids to deal with geometry approximation errors. Usually, the problem geometry is represented in CAD by means of NURBS or generalized tensorial NURBS like T-splines or hierarchical splines. This concept, originally invented for the finite element method (IGAFEM) has proved very fruitful in applications. Since CAD directly provides a parametrization of the boundary, this makes the BEM the most attractive numerical scheme, if applicable (i.e., provided that the fundamental solution of the differential operator is explicitly known), since the non-trivial meshing of the volume is avoided. The proposed research aims at the mathematical foundation of adaptive isogeometric BEM (IGABEM) for second-order elliptic PDEs with focus on convergence and optimal rates: First, the derivation and numerical analysis of a posteriori error estimates for 2D and 3D weakly-singular and hyper-singular integral equations which are suitable for IGABEM. This also includes the detection of smoothness properties of the (unknown) exact solution. Second, the development of adaptive algorithms for IGABEM. Unlike standard adaptive strategies for FEM and BEM with piecewise polynomials, the developed strategies will monitor and steer the h-refinement of the underlying mesh as well as the local smoothness properties of the IGABEM ansatz functions. The goal is that the adaptive algorithm detects singularities of the unknown solution which are resolved by h-refinement, as well as possible jumps (for weakly-singular integral equations), while high smoothness of the ansatz functions is automatically enforced, where the solution appears to be smooth. Compared to standard BEM and FEM, this will allow to reduce the number of degrees of freedom, while preserving the same convergence behavior of the overall scheme. In particular, we expect to improve the pre-asymptotic behavior of the numerical discretization. Third, we aim for a thorough convergence and quasi-optimality analysis for the developed adaptive strategies. Unlike the available concepts in the literature, one further challenge is that the discrete IGABEM ansatz spaces will not be nested if the algorithm increases the local smoothness properties at old knots. Finally, all theoretical findings will be implemented and provided to the academic public to underline the practical impact of the developed mathematical concepts and results.

The ultimate goal of any numerical scheme is to compute a discrete solution with error below a prescribed tolerance at the expense of, up to a multiplicative constant, the minimal computational cost. In computational PDEs, the convergence behavior of numerical schemes is, however, usually spoiled by singularities of the given data and/or the unknown solution. One remedy is to use adaptive strategies. Although adaptive strategies are successfully employed since the eighties, their mathematical understanding is still in its infancy and restricted to model problems and standard methods like the finite element method (FEM) or the boundary element method (BEM) with piecewise polynomials of fixed polynomial order. The central idea of isogeometric analysis (IGA) is to use the same ansatz functions for the discretization of the PDE at hand as are used for the representation of the problem geometry, which in particular avoids to deal with geometry approximation errors. Usually, the problem geometry is represented in CAD by means of NURBS or generalized tensorial NURBS like T-splines or hierarchical splines. This concept, originally invented for the finite element method (IGAFEM) has proved very fruitful in applications. Since CAD directly provides a parametrization of the boundary, this makes the BEM the most attractive numerical scheme, if applicable (i.e., provided that the fundamental solution of the differential operator is explicitly known), since the non-trivial meshing of the volume is avoided. For IGA-based discretizations, the project developed and analyzed adaptive algorithms, which refine the underlying meshes as well as the local smoothness of the NURBS ansatz functions such that the error between the unknown exact solution and the computed IGA solutions decay with the best possible convergence rates (with respect to the degrees of freedom). Furthermore, we have developed the mathematical framework to understand optimal convergence rates with respect to the computational costs (and hence the overall computational time).