A Nitsche-based domain decomposition method for hypersingular integral equations

We introduce and analyze a Nitsche-based domain decomposition method for the solution of hypersingular integral equations. This method allows for discretizations with non-matching grids without the necessity of a Lagrangian multiplier, as opposed to the traditional mortar method. We prove its almost quasi-optimal convergence and underline the theory by a numerical experiment.Comment: 21 pages, 5 figure

Block-adaptive Cross Approximation of Discrete Integral Operators

In this article we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the block-wise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Local convergence of the FEM for the integral fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm

Random Vibrations of Nonlinear Continua Endowed with Fractional Derivative Elements

In this paper, two techniques are proposed for determining the large displacement statistics of random exciting continua endowed with fractional derivative elements: Boundary Element Method (BEM) based Monte Carlo simulation; and Statistical Linearization (SL). The techniques are applied to the problem of nonlinear beam and plate random response determination in the case of colored random external load. The BEM is implemented in conjunction with a Newmark scheme for estimating the system response in the time domain in conjunction with repeated simulations, while SL is used for estimating efficiently and directly, albeit iteratively, the response statistics