In this article we study discontinuous Galerkin finite element discretizations of linearsecond-order elliptic partial differential equations with Dirac delta right-hand side. Inparticular, assuming that the underlying computational mesh is quasi-uniform, we derive ana priori bound on the error measured in terms of theL2-norm. Additionally, we develop residual-based aposteriori error estimators that can be used within an adaptive mesh refinementframework. Numerical examples for the symmetric interior penalty scheme are presentedwhich confirm the theoretical results.

We present a heterogeneous finite element method for the solution of a high-dimensionalpopulation balance equation, which depends both the physical and the internal propertycoordinates. The proposed scheme tackles the two main difficulties in the finite elementsolution of population balance equation: (i) spatial discretization with the standardfinite elements, when the dimension of the equation is more than three, (ii) spuriousoscillations in the solution induced by standard Galerkin approximation due to pureadvection in the internal property coordinates. The key idea is to split thehigh-dimensional population balance equation into two low-dimensional equations, anddiscretize the low-dimensional equations separately. In the proposed splitting scheme, theshape of the physical domain can be arbitrary, and different discretizations can beapplied to the low-dimensional equations. In particular, we discretize the physical andinternal spaces with the standard Galerkin and Streamline Upwind Petrov Galerkin (SUPG)finite elements, respectively. The stability and error estimates of the Galerkin/SUPGfinite element discretization of the population balance equation are derived. It is shownthat a slightly more regularity, i.e.the mixed partial derivatives of the solution has to be bounded, is necessary for theoptimal order of convergence. Numerical results are presented to support the analysis.

Since matrix compression has paved the way for discretizing the boundary integralequation formulations of electromagnetics scattering on very fine meshes, preconditionersfor the resulting linear systems have become key to efficient simulations. Operatorpreconditioning based on Calderón identities has proved to be a powerful device fordevising preconditioners. However, this is not possible for the usual first-kind boundaryformulations for electromagnetic scattering at general penetrable composite obstacles. Wepropose a new first-kind boundary integral equation formulation following the reasoningemployed in [X. Clayes and R. Hiptmair, Report 2011-45, SAM, ETH Zürich (2011)] foracoustic scattering. We call it multi-trace formulation, because itsunknowns are two pairs of traces on interfaces in the interior of the scatterer. We give acomprehensive analysis culminating in a proof of coercivity, and uniqueness and existenceof solution. We establish a Calderón identity for the multi-trace formulation, which formsthe foundation for operator preconditioning in the case of conforming Galerkin boundaryelement discretization.

In this paper we study optimal control computation based on the control parameterization method for a class of optimal control problems involving nonlinear systems with multiple time delays subject to continuous state inequality constraints. Both the state and the control are allowed to have different time delays, and they are uncorrelated in this system. The control of the dynamical system is approximated by a piecewise constant function whose heights are taken as decision vectors. The formulae for computing the gradients of the cost and constraint functions are then derived. Based on this, a computational method for finding the optimal control is developed by utilizing the Sequential Quadratic Programming (SQP) algorithm with an active set strategy. The computational method is applied to an industrial problem arising in the purification process of zinc hydrometallurgy. Numerical simulation shows that the amount of zinc powder that is needed can be decreased significantly, thus avoiding wastage of resources.

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

We study the one-dimensional conservation law. We use a characteristic surface to define a class of functions, within which the integral version of the conservation law is solved in a simple and direct way. A simple algorithm for computing the unique solution is developed. The method uses the equal-area principle and yields the solution for any given time directly.

In this paper, a lower bound is established for the local energy of partial sum of eigenfunctions for Laplace-Beltrami operators (in Riemannian manifolds with low regularity data) with general boundary condition. This result is a consequence of a new pointwise and weighted estimate for Laplace-Beltrami operators, a construction of some nonnegative function with arbitrary given critical point location in the manifold, and also two interpolation results for solutions of elliptic equations with lateral Robin boundary conditions.

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.