Clemens Hofreither

A robust multigrid method for Isogeometric Analysis in two dimensions using boundary correction

We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers showing optimal convergence behavior. However, the naive application of multigrid to the isogeometric case results in significant deterioration of the convergence rates if the spline degree is increased. Recently, a robust approximation error estimate and a corresponding inverse inequality for B-splines of maximum smoothness have been shown, both with constants independent of the spline degree. We use these results to construct multigrid solvers for discretizations of two-dimensional problems based on tensor product B-splines with maximum smoothness which exhibit robust convergence rates.

Robust Multigrid for Isogeometric Analysis Based on Stable Splittings of Spline Spaces

We present a robust and efficient multigrid method for single-patch isogeometric discretizations using tensor product B-splines of maximum smoothness. Our method is based on a stable splitting of the spline space into a large subspace of “interior” splines which satisfy a robust inverse inequality, as well as one or several smaller subspaces which capture the boundary effects responsible for the spectral outliers which occur in isogeometric analysis. We then construct a multigrid smoother based on an additive subspace correction approach, applying a different smoother to each of the subspaces. For the interior splines, we use a mass smoother, whereas the remaining components are treated with suitably chosen Kronecker product smoothers or direct solvers. We prove that the resulting multigrid method exhibits iteration numbers which are robust with respect to the spline degree and the mesh size. Furthermore, it can be efficiently realized for discretizations of problems in arbitrarily high geometric dimensio...

Interpolation of harmonic functions based on Radon projections

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr’s formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

Convection‐adapted BEM‐based FEM

We present a new discretization method for homogeneous convection-diffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

A Non‐standard Finite Element Method for Convection‐Diffusion‐Reaction Problems on Polyhedral Meshes

We present a nonstandard finite element method for elliptic partial differential equations with piecewise constant coefficients which is based on element‐local boundary integral operators. The method is able to treat general polyhedral meshes and employs locally PDE‐harmonic trial functions. In this paper, we apply this method for the first time to convection‐diffusion‐reaction problems. We assume that the coefficients of the PDE are elementwise constant such that a fundamental solution is available locally.We perform a preliminary numerical study of the method for convection‐dominated problems in three dimensions and compare its performance for this application to a standard Finite Element Method. The results show promising stability properties.

Harmonic interpolation based on Radon projections along the sides of regular polygons

Given information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

On Full Multigrid Schemes for Isogeometric Analysis

We investigate a geometric full multigrid method for solving the large sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. We observe that the full multigrid approach performs much better than the V-cycle multigrid method in many cases, in particular in higher dimensions with increased spline degrees. Often, a single cycle of the full multigrid process is sufficient to obtain a quasi-optimal solution in the L 2-norm. A modest increase in the number of smoothing steps suffices to restore optimality in cases where the V-cycle performs badly.

New results on regularity and errors of harmonic interpolation using Radon projections

We study interpolation of harmonic functions in the unit disk with a finite number of values of the Radon projection along prescribed chords as the input data. We seek the interpolant in the space of harmonic polynomials in such a way that it matches the given projection values exactly. In this setting, we investigate schemes where all chords are divided into two sets of parallel chords. We give necessary and sufficient conditions for a scheme of this type to result in a uniquely solvable interpolation problem. As a second new result, we generalize the previously known error estimates for schemes with equispaced chord angles, both to allow for a larger class of chord choices and to obtain new error estimates in fractional Sobolev norms.

Spectral Analysis of Geometric Multigrid Methods for Isogeometric Analysis

We investigate geometric multigrid methods for solving the large, sparse linear systems which arise in isogeometric discretizations of elliptic partial differential equations. We observe that the performance of standard V-cycle iteration is highly dependent on the spatial dimension as well as the spline degree of the discretization space. Conjugate gradient iteration preconditioned with one V-cycle mitigates this dependence, but does not eliminate it. We perform both classical local Fourier analysis as well as a numerical spectral analysis of the two-grid method to gain better understanding of the underlying problems and observe that classical smoothers do not perform well in the isogeometric setting.

A Rigorous Error Analysis of Coupled FEM-BEM Problems with Arbitrary Many Subdomains

In this article, we provide a rigorous a priori error estimate for the symmetric coupling of the finite and boundary element method for the potential problem in three dimensions. Our theoretical framework allows an arbitrary number of polyhedral subdomains. Our bound is not only explicit in the mesh parameter, but also in the subdomains themselves: the bound is independent of the number of subdomains and involves only the shape regularity constants of a certain coarse triangulation aligned with the subdomain decomposition. The analysis includes the so-called BEM-based FEM as a limit case.