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. AbstractAn enriched partition-of-unity (PU) finite element method is developed to solve timedependent diffusion problems. In the present PU formulation, an exponential solution describing the spatial diffusion decay is embedded in the finite element shape function. It results in an enriched approximation, which is in the form of local asymptotic expansion. The temporal decay in the solution is embedded naturally in the PU expansion so that, unlike previous works in this area, the same system matrices may be used for every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more robust and efficient, and yields more accurate solutions for a prescribed number of degrees of freedom. On the other hand, the notorious difficulty encountered in the meshless method in satisfying the essential boundary conditions is circumvented. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analysed on two applications: the transient heat equation with a single source and with multiple sources. The aim of such a method compared to the classical finite element method is to solve time-dependent diffusion applications efficiently and with an appropriate level of accuracy.

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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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