It is well known that Galerkin finite element methods suffer from pollution error when solving wave problems. To reduce the pollution impact on the solution different approaches were proposed to enrich the finite element method with wave-like functions so that the exact wavenumber is incorporated into the finite element approximation space. Solving wave problems with isogeometric analysis was also investigated in the literature where the superior behaviour of isogeometric analysis due to higher continuity in the underlying basis has been studied. Recently, a plane wave enriched isogeometric analysis was introduced for acoustic problems. However, it remains unquantified the impact of these different approaches on the pollution or how they perform compared to each other. In this work, we show that isogeometric analysis outperforms finite element method in dealing with pollution. We observe similar behaviour when both the methods are enriched with plane waves. Using higher order polynomials with fewer enrichment functions seems to improve the pollution compared to lower order polynomials with more functions. However, the latter still leads to smaller errors using similar number of degrees of freedom. In conclusion, we propose that partition of unity isogeometric analysis can be an efficient tool for wave problems as enrichment eliminates the need for domain re-meshing at higher frequencies and also due to its ability to capture the exact geometry even on coarse meshes as well as its improved pollution behaviour.

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractProblems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media. These problems are usually very demanding in terms of computations with the conventional finite element method. On the other hand different enriched finite element methods such as the partition of unity which proved to be very successful in treating similar problems, are developed and studied for homogeneous media. In this work we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials. The idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method. Although presented within the partition of unity context the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials.

Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. SUMMARYThe Partition of Unity Method has become an attractive approach for extending the allowable frequency range for wave simulations beyond that available using piecewise polynomial elements. The non-uniqueness of solution obtained from the Conventional Boundary Integral Equation (CBIE) is well known. The CBIE derived through Green's identities suffers from a problem of non-uniqueness at certain characteristic frequencies. Two of the standard methods of overcoming this problem are the so-called CHIEF method and that of Burton and Miller. The latter method introduces a hypersingular integral which may be treated in various ways. In this paper we present the collocation Partition of Unity Boundary Element Method (PUBEM) for the Helmholtz problem and compare the performance of CHIEF against a Burton-Miller formulation regularised using the approach of Li and Huang.

A virtual element method for the quasilinear equation $-\textrm {div} ({\boldsymbol \kappa }(u)\operatorname {grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.