Goal oriented error control for stationary incompressible flow coupled to a heat equation

In this work, we apply goal oriented error estimation to a stationary Navier-Stokes benchmark problem coupled with the heat equation. Furthermore, we compare three different methods for the sensitivity weight recovery

Recursion relations for hp-FEM Element Matrices on quadrilaterals

In this paper we consider higher order shape functions for finite elements on quadrilaterals. Using tensor products of suitable Jacobi polynomials, it can be proved that the corresponding mass and stiffness matrices are sparse with respect to polynomial degree p . Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffness matrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to compute the remaining nonzero entries in optimal arithmetic complexity. Besides the H1 case also the conformal basis functions for H(Div) and H(Curl) are investigated

Algorithmic realization of the solution to the sign conflict problem for hanging nodes on hp-hexahedral N\'ed\'elec elements

While working with N\'ed\'elec elements on adaptively refined meshes with hanging nodes, the orientation of the hanging edges and faces must be taken into account. Indeed, for non-orientable meshes, there was no solution and implementation available to date. The problem statement and corresponding algorithms are described in great detail. As a model problem, the time-harmonic Maxwell's equations are adopted because N\'ed\'elec elements constitute their natural discretization. The implementation is performed within the finite element library deal.II. The algorithms and implementation are demonstrated through four numerical examples on different uniformly and adaptively refined meshes

Sparsity optimized high order finite element functions for H(curl) on tetrahedra

AbstractH(curl) conforming finite element discretizations are a powerful tool for the numerical solution of the system of Maxwellʼs equations in electrodynamics. In this paper we construct a basis for conforming high-order finite element discretizations of the function space H(curl) in 3 dimensions. We introduce a set of hierarchic basis functions on tetrahedra with the property that both the L2-inner product and the H(curl)-inner product are sparse with respect to the polynomial degree. The construction relies on a tensor-product based structure with properly weighted Jacobi polynomials as well as an explicit splitting of the basis functions into gradient and non-gradient functions. The basis functions yield a sparse system matrix with O(1) nonzero entries per row.The proof of the sparsity result on general tetrahedra defined in terms of their barycentric coordinates is carried out by an algorithm that we implemented in Mathematica. A rewriting procedure is used to explicitly evaluate the inner products. The precomputed matrix entries in this general form for the cell-based basis functions are available online

Multigoal-oriented a posteriori error control for heated material processing using a generalized Boussinesq model

In this work, we develop a posteriori error control for a generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier–Stokes equations are coupled with a stationary heat equation. The coupled problem is modeled and solved in a monolithic fashion. The focus is on multigoal-oriented error estimation with the dual-weighted residual method in which an adjoint problem is utilized to obtain sensitivity measures with respect to several goal functionals. The error localization is achieved with the help of a partition-of-unity in a weak formulation, which is specifically convenient for coupled problems as we have at hand. The error indicators are used to employ adaptive algorithms, which are substantiated with several numerical tests such as one benchmark and two further experiments that are motivated from laser material processing. Therein, error reductions and effectivity indices are consulted to establish the robustness and efficiency of our framework

A residual‐based error estimator and mesh adaptivity for the time harmonic Maxwell equations applied to a Y‐beam splitter

In this work, local mesh adaptivity for the time harmonic Maxwell equations is studied. The main purpose is to apply a known a posteriori residual-based error estimator from the literature and to investigate its performance for a Y-beam splitter setting. This configuration is an important prototype for the design of optical systems within the excellence cluster PhoenixD. Specifically, the branching region is of interest and requires a high accuracy of the numerical simulation. One numerical example shows the performance of our approach

Hierarchical LU preconditioning for the time-harmonic Maxwell equations

The time-harmonic Maxwell equations are used to study the effect of electric and magnetic fields on each other. Although the linear systems resulting from solving this system using FEMs are sparse, direct solvers cannot reach the linear complexity. In fact, due to the indefinite system matrix, iterative solvers suffer from slow convergence. In this work, we study the effect of using the inverse of $\mathcal{H}$-matrix approximations of the Galerkin matrices arising from N\'ed\'elec's edge FEM discretization to solve the linear system directly. We also investigate the impact of applying an $\mathcal{H}-LU$ factorization as a preconditioner and we study the number of iterations to solve the linear system using iterative solvers