## Two comments on filtering (artificial viscosity) for Chebyshev and Legendre spectral and spectral element methods: Preserving boundary conditions and interpretation of the filter as a diffusion.(English)Zbl 0920.65046

Applying the strategy for reducing numerical noise with a spectral method by filtering the coefficients, i.e. by replacing the truncated series $$u_N(x)=\sum_{j=0}^Na_jT_j(x)$$ by its filtrate $$u_F(x;N)=\sum_{j=0}^Na_j\sigma(j/N)T_j(x)$$, for some filter function $$\sigma$$, the boundary conditions can be modified. The note presents a modification to filtering so that $$u_F(x)$$ satisfies the same boundary conditions as $$u_N(x)$$. The key idea is to rewrite $$u_N(x)$$ in terms of new basis functions which individually satisfy homogeneous boundary conditions and then apply the filter to modify the coefficients of the new expansion. The filtered sum is then converted back into the original basis.

### MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations
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### References:

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