We propose a method that combines isogeometric analysis(IGA)with the discontinuous Galerkin(DG)method for solving elliptic equations on 3-dimensional(3D)surfaces consisting of multiple patches.DG ideology is adopted across the patch interfaces to glue the multiple patches,while the traditional IGA,which is very suitable for solving partial differential equations(PDEs)on(3D)surfaces,is employed within each patch.Our method takes advantage of both IGA and the DG method.Firstly,the time-consuming steps in mesh generation process in traditional finite element analysis(FEA)are no longer necessary and refinements,including h-refinement and p-refinement which both maintain the original geometry,can be easily performed by knot insertion and order-elevation(Farin,in Curves and surfaces for CAGD,2002).Secondly,our method can easily handle the cases with non-conforming patches and different degrees across the patches.Moreover,due to the geometric flexibility of IGA basis functions,especially the use of multiple patches,we can get more accurate modeling of more complex surfaces.Thus,the geometrical error is significantly reduced and it is,in particular,eliminated for all conic sections.Finally,this method can be easily formulated and implemented.We generally describe the problem and then present our primal formulation.A new ideology,which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain(the former is easier to get),is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form,and the error analysis under both the DG norm and the L2 norm.The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L2 norm.Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.
LSQR, a Lanczos bidiagonalization based Krylov subspace iterative method, and its mathematically equivalent conjugate gradient for least squares problems(CGLS) applied to normal equations system, are commonly used for large-scale discrete ill-posed problems. It is well known that LSQR and CGLS have regularizing effects, where the number of iterations plays the role of the regularization parameter. However, it has long been unknown whether the regularizing effects are good enough to find best possible regularized solutions. Here a best possible regularized solution means that it is at least as accurate as the best regularized solution obtained by the truncated singular value decomposition(TSVD) method. We establish bounds for the distance between the k-dimensional Krylov subspace and the k-dimensional dominant right singular space. They show that the Krylov subspace captures the dominant right singular space better for severely and moderately ill-posed problems than for mildly ill-posed problems. Our general conclusions are that LSQR has better regularizing effects for the first two kinds of problems than for the third kind, and a hybrid LSQR with additional regularization is generally needed for mildly ill-posed problems. Exploiting the established bounds, we derive an estimate for the accuracy of the rank k approximation generated by Lanczos bidiagonalization. Numerical experiments illustrate that the regularizing effects of LSQR are good enough to compute best possible regularized solutions for severely and moderately ill-posed problems, stronger than our theory, but they are not for mildly ill-posed problems and additional regularization is needed.
