Chebyshev interpolation for nonlinear eigenvalue problems

This work is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems and typically exclude the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. Our approach is intended for situations where the eigenvalues of interest are located on the real line or, more generally, on a pre-specified curve in the complex plane. A first-order perturbation analysis for nonlinear eigenvalue problems is performed. Combined with an approximation result for Chebyshev interpolation, this shows exponential convergence of the obtained eigenvalue approximations with respect to the degree of the approximating polynomial. Preliminary numerical experiments demonstrate the viability of the approach in the context of boundary element methods.

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems

Assyr Abdulle, Gilles Vilmart

A fully discrete analysis of the finite element heterogeneous multiscale method for a class of nonlinear elliptic homogenization problems of nonmonotone type is proposed. In contrast to previous results obtained for such problems in dimension

d\leq2

for the

H^1

norm and for a semi-discrete formulation [W.E, P. Ming and P. Zhang, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156], we obtain optimal convergence results for dimension

d\leq3

and for a fully discrete method, which takes into account the microscale discretization. In addition, our results are also valid for quadrilateral finite elements, optimal a-priori error estimates are obtained for the

H^1

and

L^2

norms, improved estimates are obtained for the resonance error and the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems.

Amer Mathematical Soc2014

MATHICSE Technical Report : Isogeometric analysis of geometric partial differential equations

Andrea Bartezzaghi, Alfio Quarteroni

We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L2-gradient ow. We analyze two energy functionals: the area, which leads to the mean curvature ow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore ow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch NURBS and discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method. To approximate the high order geometric PDEs we use high order continuous NURBS basis functions. Instead, for the time discretization of the nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) with extrapolation of the geometric quantities involved in the weak formulation of the problem; in this manner, we solve a linear problem at each time step. We report numerical results concerning the mean curvature and Willmore ows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme.

MATHICSE2016