Mixed methods for degenerate elliptic problems and application to fractional Laplacian

Cejas, María E.; Durán, Ricardo G.; Prieto, Mariana I.

doi:10.1051/m2an/2020068

Cejas, María E. ; Durán, Ricardo G. ; Prieto, Mariana I.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S993-S1019

Résumé

We analyze the approximation by mixed finite element methods of solutions of equations of the form −div (a∇u) = g, where the coefficient a = a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient a belongs to the Muckenhoupt class A₂. The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart–Thomas spaces of arbitrary order, obtaining optimal order error estimates for simplicial elements in any dimension and for convex quadrilateral elements in the two dimensional case, in both cases under a regularity assumption on the family of meshes. For the lowest order case we show that the regularity assumption can be removed and prove anisotropic error estimates which are of interest in problems with boundary layers. Finally we apply the results to a problem arising in the solution of the fractional Laplace equation.

MR Zbl

DOI : 10.1051/m2an/2020068

Classification : 65N30, 35J70
Keywords: Mixed finite elements, degenerate elliptic problems, fractional Laplacian

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     author = {Cejas, Mar{\'\i}a E. and Dur\'an, Ricardo G. and Prieto, Mariana I.},
     title = {Mixed methods for degenerate elliptic problems and application to fractional {Laplacian}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {S993--S1019},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {Suppl\'ement},
     doi = {10.1051/m2an/2020068},
     mrnumber = {4221331},
     zbl = {1477.65200},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2020068/}
}

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Cejas, María E.; Durán, Ricardo G.; Prieto, Mariana I. Mixed methods for degenerate elliptic problems and application to fractional Laplacian. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S993-S1019. doi: 10.1051/m2an/2020068

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