L. Beirão da Veiga

ISOGEOMETRIC COLLOCATION METHODS

We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

The Hitchhiker's Guide to the Virtual Element Method

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

Mathematical analysis of variational isogeometric methods

This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.

Virtual Element Method for general second-order elliptic problems on polygonal meshes

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the L2-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes

We develop and analyze a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form. These methods are derived from the local consistency condition that is exact for polynomials of any degree

Mimetic finite difference method for the Stokes problem on polygonal meshes

m\geq1

ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES

. The degrees of freedom are (a) solution values at the quadrature nodes of the Gauss-Lobatto formulas on each mesh edge, and (b) solution moments inside polygons. The convergence of the method is proven theoretically and an optimal error estimate is derived in a mesh-dependent norm that mimics the energy norm. Numerical experiments confirm the convergence rate that is expected from the theory.

An isogeometric method for the Reissner–Mindlin plate bending problem

Various approaches to extend finite element methods to non-traditional elements (general polygons, pyramids, polyhedra, etc.) have been developed over the last decade. The construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis. The mimetic finite difference (MFD) method works on general polygonal meshes and has many similarities with low-order finite element methods. Both schemes try to preserve the fundamental properties of the underlying physical and mathematical models. The essential difference between the two schemes is that the MFD method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices. Since no extension of basis functions inside the mesh elements is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex elements. In this article, we present a new MFD method for the Stokes problem on arbitrary polygonal meshes and analyze its stability. The method is developed for the general case of tensor coefficients, which allows us to apply it to a linear elasticity problem, as well. Numerical experiments show, for the velocity variable, second-order convergence in a discrete L^2 norm and first-order convergence in a discrete H^1 norm. For the pressure variable, first-order convergence is shown in the L^2 norm.

BDDC PRECONDITIONERS FOR ISOGEOMETRIC ANALYSIS

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirao da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

Convergence analysis of the high-order mimetic finite difference method

We present a new isogeometric method for the discretization of the Reissner–Mindlin plate bending problem. The proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections Wh and a space of smooth discrete rotations Θh such that the Kirchhoff constraint is exactly satisfied at the limit. Therefore we obtain a formulation which is natural from the theoretical/mechanical viewpoint and locking-free by construction. We prove that the method is uniformly stable and satisfies optimal convergence estimates. Finally, the theoretical results are fully supported by numerical tests.