Giancarlo Sangalli

Isogeometric Discrete Differential Forms in Three Dimensions

The concept of isogeometric analysis (IGA) was first applied to the approximation of Maxwell equations in [A. Buffa, G. Sangalli, and R. Vazquez, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1143-1152]. The method is based on the construction of suitable B-spline spaces such that they verify a De Rham diagram. Its main advantages are that the geometry is described exactly with few elements, and the computed solutions are smoother than those provided by finite elements. In this paper we develop the theoretical background to the approximation of vector fields in IGA. The key point of our analysis is the definition of suitable projectors that render the diagram commutative. The theory is then applied to the numerical approximation of Maxwell source problems and eigenproblems, and numerical results showing the good behavior of the scheme are also presented.

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

Summary.In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

LINK-CUTTING BUBBLES FOR THE STABILIZATION OF CONVECTION-DIFFUSION-REACTION PROBLEMS

It is known that the addition and elimination of suitable bubble functions can result in a stabilized scheme of the SUPG-type. Residual-Free Bubbles (RFB), in particular, can assure a quasi-optimal stabilized scheme, but they are difficult to compute in one dimension and nearly impossible to compute in two and three dimensions, unless in special limit cases. Strongly convection-dominated problems (without reaction terms) are one of these cases, where it is possible to find reasonably simple computable bubbles that provide a stabilizing effect as good as that of true RFB. Here, although in a one-dimensional framework, we analyze the case in which a non-negligible reaction term is present, and provide a simple recipe for spotting a suitable bubble space (adding two bubbles to each element) that provides a very good stabilizing effect. The method adapts very well to all regimes with continuous transitions from one regime to another. It is clear that the one-dimensional case, in itself, has no real interest. We believe, however, that the discussion can cast some light on the interaction between convection and reaction that could be useful in future works dealing with multidimensional, more realistic problems.

Analysis of a Multiscale Discontinuous Galerkin Method for Convection-Diffusion Problems

We study a multiscale discontinuous Galerkin method introduced in [T. J. R. Hughes, G. Scovazzi, P. Bochev, and A. Buffa, Comput. Meth. Appl. Mech. Engrg., 195 (2006), pp. 2761-2787] that reduces the computational complexity of the discontinuous Galerkin method, seemingly without adversely affecting the quality of results. For a stabilized variant we are able to obtain the same error estimates for the convection-diffusion equation as for the usual discontinuous Galerkin method. We assess the stability of the unstabilized case numerically and find that the inf-sup constant is positive, bounded uniformly away from zero, and very similar to that for the usual discontinuous Galerkin method.

Accurate, efficient, and (iso)geometrically flexible collocation methods for phase-field models

We propose new collocation methods for phase-field models. Our algorithms are based on isogeometric analysis, a new technology that makes use of functions from computational geometry, such as, for example, Non-Uniform Rational B-Splines (NURBS). NURBS exhibit excellent approximability and controllable global smoothness, and can represent exactly most geometries encapsulated in Computer Aided Design (CAD) models. These attributes permitted us to derive accurate, efficient, and geometrically flexible collocation methods for phase-field models. The performance of our method is demonstrated by several numerical examples of phase separation modeled by the Cahn-Hilliard equation. We feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo-spectral collocation methods, and is a viable alternative to classical collocation methods.

Global and Local Error Analysis for the Residual-Free Bubbles Method Applied to Advection-Dominated Problems

We prove the stability and a priori global and local error analysis for the residual-free bubbles (RFB) finite element method applied to advection-dominated advection-diffusion problems.

ROBUST A-POSTERIORI ESTIMATOR FOR ADVECTION-DIFFUSION-REACTION PROBLEMS

We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem. The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a √log(Pe) factor, where Pe is the global Peclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature. We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations

In this paper we introduce methods for electromagnetic wave propagation, based on splines and on T-splines. We define spline spaces which form a De Rham complex and following the isogeometric paradigm, we map them on domains which are (piecewise) spline or NURBS geometries. We analyze their geometric and topological structure, as related to the connectivity of the underlying mesh, and we present degrees of freedom together with their physical interpretation. The theory is then extended to the case of meshes with T-junctions, leveraging on the recent theory of T-splines. The use of T-splines enhance our spline methods with local refinement capability and numerical tests show the efficiency and the accuracy of the techniques we propose.

Fast formation of isogeometric Galerkin matrices by weighted quadrature

Abstract In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for p large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree k -refinement in isogeometric analysis.

Analysis-suitable G1 multi-patch parametrizations for C1 isogeometric spaces

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to C p - 1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h-refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently (Kapl et al., 2015b) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. We study h-refinement for C 1 continuous isogeometric spaces over multi-patch domains.We introduce analysis-suitable G 1 (AS G 1 ) geometry parametrizations.AS G 1 parametrizations allow optimal approximation properties.For non-AS G 1 geometries the solution may be locked and convergence is prevented.