Numerical Analysis Seminar

In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree p on each mesh element. Earlier works showed that there is a \trial-to-test" operator T, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator T is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply T. In practical computations, T is approximated using polynomials of some degree r > p on each mesh element. We show that this approximation maintains optimal convergence rates, provided that r p + N, where N is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.

Fast computation of drum modes using the spectrum of the Neumann-to-Dirichlet map We present and analyze a new method for numerical computation of the spectrum and eigenfunctions of a planar star-shaped domain with Dirichlet boundary condition. The method is 'fast' since it is computes a cluster of eigenfunctions (numbering of order the square-root of the eigenvalue) in the time usually taken to compute a single one. In practice, with 400 wavelengths across the domain, and relative error 1e-10, this speed-up is around 1e3. It is related to the little-understood 'scaling method', but, in contrast, has a rigorous error analysis and allows higher-order accuracy. We will include some applications to quantum chaos. Joint work with Andrew Hassell (ANU).

Standard finite element error analysis for elliptic problems is based on the Cea and Bramble-Hilbert lemmas in combination with transformation properties of Sobolev norms. Boundary element analysis is no different in this respect, main additional ingredients are the mapping properties of boundary integral operators.

We discuss the following problem: Given a target function $u^* \in L^2 (\Omega)$, what should be the Neumann data $\lambda^*$ so that its harmonic extension $u_{\lambda^*}$ into $\Omega$ is the closest function to $u^*$ in the $L_2(\Omega)$ norm. For convex polygonal domain, we show that regularization is not needed if the control space for the Neumann data is chosen properly. In the second part of the talk we discuss robust solvers for the discrete Hessian system posed on the boundary of $\Omega$.

HDG for Stokes flows have been recently devised which provide optimally convergent approximations for the velocity gradient, the velocity and the pressure: When polynomials of degree k are used, these approximations converge with order k+1. Moreover, a local postprocessing provides a globally divergence-free approximation for the velocity which convergences with order 1 for k=0 and with order k+2 otherwise. We show that these properties remain unchanged when the stabilization function associated to the interelement jump of the normal component of the velocity is "sent to infinity" to obtain an H(div)-conforming HDG method.

We propose a projection-based a priori error analysis of finite element methods for second-order elliptic problems. The analysis is unifying because it applies to a large class of methods including the hybridized version of most well-known mixed methods as well as several hybridizable discontinuous Galerkin (HDG) methods. The novelty of the approach is that it reduces the whole error analysis to the element-by-element construction of an auxiliary projection satisfying certain orthogonality and approximation properties, and to the verification of very simple inclusion properties of the local spaces defining the methods. We extend this approach to Stokes equations and linear elasticity.

Equations describing flow through heterogeneous porous media are often difficult to solve by direct numerical simulation due to their expensive computational requirement. In this talk, numerical convergence results will be presented for solving the second order elliptic equation with variable coefficient by using two low-computational cost methods that apply analytical results incorporated into multigrid. Besides the methods being inheritably fast, the new feature is that the variable coefficient can describe a general non-periodic media. The first approach defines the operator at each coarse-scale of the v-cycle using an analytical approximation of the upscale tensor, in the same spirit as iterative homogenization procedures. The second approach, which may be termed as aggressive coarsening, obtains the solution by skipping few levels in the v-cycle. It is obtained as an extension of the first approach by further defining a multiscale prolongation operator by using an analytical approximation of the solution of the well known cell-problem from homogenization theory. The talk will concentrate on demonstrating numerically that convergence can be achieved, using typical implementation procedures of the multigrid method, meaning standard Gauss-Seidel smoothing and interpolation and restriction procedures. For the first method, the robustness of the algorithm is measured by comparing it with the algorithm using arithmetic and harmonic averages of the coefficient. For the second method, the comparison is done by using its respective numerical counterpart. The examples illustrate the application for media having separable scales, such as periodic, non-periodic and homogeneous random fields and, the SPE10 benchmark project as an application of non-separable scale.