Martin Campos Pinto

Kolja Brix, Martin Campos Pinto, Claudio Canuto et al.

This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of the condition numbers that is logarithmic in the polynomial degrees when all degrees are equal and quadratic otherwise, our main objective is to realize full robustness with respect to arbitrarily large locally varying polynomial degrees degrees, i.e., under mild grading constraints condition numbers stay uniformly bounded with respect to the mesh size and variable degrees. The conceptual foundation of the envisaged preconditioners is the auxiliary space method. The main conceptual ingredients that will be shown in this framework to yield "optimal" preconditioners in the above sense are Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic nested dyadic grids as well as specially adapted wavelet preconditioners for the resulting low order auxiliary problems. Moreover, the preconditioners have a modular form that facilitates somewhat simplified partial realizations. One of the components can, for instance, be conveniently combined with domain decomposition, at the expense though of a logarithmic growth of condition numbers. Our analysis is complemented by quantitative experimental studies of the main components.

Martin Campos Pinto

We present a new class of particle methods with deformable shapes that converge in the uniform norm without requiring remappings, extended overlapping or vanishing moments for the particles. The crux of the method is to use polynomial expansions of the backward characteristic flow to transport the numerical particles with improved accuracy: in the first order case the method consists of representing the transported density with linearly-transformed particles, the second order version computes quadratically-transformed particles, and so on. For programming purposes we provide explicit implementations of the resulting LTP and QTP schemes that only involve pointwise evaluations of the forward characteristic flow, and also come with local indicators for the accuracy of the corresponding transport scheme. Numerical tests using different transport problems demonstrate the accuracy of the proposed methods compared to standard particle schemes, and establish their robustness with respect to the remapping period. In particular, it is shown that QTP particles can be transported without remappings on very long periods of times, without hampering the accuracy of the numerical solutions. Finally, a dynamic criterion is proposed to automatically select the time steps where the particles should be remapped. The strategy does not require additional inter-particle communications, and it is validated by numerical experiments.

Martin Campos Pinto, Frédérique Charles, Bruno Després et al.

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Després, Numer. Algorithms, 76(3), (2017)] and [B. Després and M. Herda, Numer. Algorithms, 77(1), (2018)] where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation.

Martin Campos Pinto, José A. Carrillo, Frédérique Charles et al.

We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in $L^1$ and $L^\infty$ norms depending on the regularity of the initial data. Moreover, we give convergence estimates in bounded Lipschitz distance for measure valued solutions. For singular interaction forces, we establish the convergence of the error between the approximated and exact flows up to the existence time of the solutions in $L^1 \cap L^p$ norm.