Markus Faustmann

We consider fractional partial differential equations posed on the full space $$\mathbb {R}^d$$ R d . Using the well-known Caffarelli-Silvestre extension to $$\mathbb {R}^d \times \mathbb {R}^+$$ R d × R + as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $$\mathbb {R}^d \times (0,\mathcal {Y})$$ R d × ( 0 , Y ) converge to the solution of the original problem as $$\mathcal {Y}\rightarrow \infty $$ Y → ∞ . Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem.

We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.

We derive and analyze a fully computable discrete scheme for fractional partial differential equations posed on the full space $\mathbb{R}^d$ . Based on a reformulation using the well-known Caffarelli-Silvestre extension, we study a modified variational formulation to obtain well-posedness of the discrete problem. Our scheme is obtained by combining a diagonalization procedure with a reformulation using boundary integral equations and a coupling of finite elements and boundary elements. For our discrete method we present a-priori estimates as well as numerical examples.

We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $\R^d \times (0,\YY)$ converge to the solution of the original problem as $\YY \rightarrow \infty$. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement. A key step hereby is an equivalence of the nodal and Scott–Zhang interpolation operators in certain weighted L 2 -norms.

The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of $${\mathcal{H}}$$H -matrices. Under a technical assumption on the mesh, we prove that root exponential convergence in the block rank can be achieved, if the block structure conforms to a standard admissibility criterion.

We consider three different methods for the coupling of the finite element method and the boundary element method, the Bielak–MacCamy coupling, the symmetric coupling, and the Johnson–Nédélec coupling. For each coupling, we provide discrete interior regularity estimates. As a consequence, we are able to prove the existence of exponentially convergent $$\mathcal {H}$$H -matrix approximants to the inverse matrices corresponding to the lowest order Galerkin discretizations of the couplings.