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Study of the enriched mixed finite element method comparing errors and computational cost with classical FEM and mixed scheme on quadrilateral meshes. (English) Zbl 1477.65206

The authors propose a new enriched version of the mixed \(H(\mathrm{div})\)-conforming finite element method and present a numerical study concerning the standard \(H^1\)-conforming finite element method, the mixed \(H(\mathrm{div})\)-conforming finite element method, and the proposed method. The methods employ quadrilateral meshes for the numerical solution of the Poisson equation on a square. Two model problems with know exact solutions are considered. In the first example, the solution is smooth, while in the second example, the solution has a steep gradient in certain regions. The authors compare the \(L^2\)-norm of the error with respect to a number of degrees of freedom and the computational cost of all three methods. The numerical results for the presented examples show that the proposed enriched mixed method of order \(k\) leads to a similar error as the mixed method of order \(k+1\), but the computational cost is significantly lower.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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