A short note on a Bernstein-Bezier basis for the pyramid

We introduce a Bernstein-Bezier basis for the pyramid, whose restriction to the face reduces to the Bernstein-Bezier basis on the triangle or quadrilateral. The basis satisfies the standard positivity and partition of unity properties common to Bernstein polynomials, and spans the same space as non-polynomial pyramid bases in the literature.Comment: Submitte

Matrix-free weighted quadrature for a computationally efficient isogeometric kk-method

The $k$-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the $k-$method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the $k$-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree $k$-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints

Total positivity and accurate computations with Gram matrices of Bernstein bases

In this paper, an accurate method to construct the bidiagonal factorization of Gram (mass) matrices of Bernstein bases of positive and negative degree is obtained and used to compute with high relative accuracy their eigenvalues, singular values and inverses. Numerical examples are included

A general approach to transforming finite elements

The use of a reference element on which a finite element basis is constructed once and mapped to each cell in a mesh greatly expedites the structure and efficiency of finite element codes. However, many famous finite elements such as Hermite, Morley, Argyris, and Bell, do not possess the kind of equivalence needed to work with a reference element in the standard way. This paper gives a generalizated approach to mapping bases for such finite elements by means of studying relationships between the finite element nodes under push-forward.Comment: 28 page