A weak Galerkin harmonic finite element method for Laplace equation. (English) Zbl 1499.65640
Summary: In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the \(P_k\)-harmonic polynomial space instead of the full polynomial space \(P_k\) is to use a much smaller number of basis functions to achieve the same accuracy when \(k\geqslant 2\). The optimal rate of convergence is derived in both \(H^1\) and \(L^2\) norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the \(P_2\)-harmonic polynomial space and using the standard \(P_2\) polynomial space are presented.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J50 | Variational methods for elliptic systems |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
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