Global estimates for mixed methods for second order elliptic equations

Authors:
Jim Douglas and Jean E. Roberts

Journal:
Math. Comp. **44** (1985), 39-52

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771029-9

MathSciNet review:
771029

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Abstract: Global error estimates in ${L^2}(\Omega )$, ${L^\infty }(\Omega )$, and ${H^{ - s}}(\Omega )$, $\Omega$ in ${{\mathbf {R}}^2}$ or ${{\mathbf {R}}^3}$, are derived for a mixed finite element method for the Dirichlet problem for the elliptic operator $Lp = - \operatorname {div}(a\;{\mathbf {grad}}\;p + {\mathbf {b}}p) + cp$ based on the Raviart-Thomas-Nedelec space ${{\mathbf {V}}_h} \times {W_h} \subset {\mathbf {H}}(\operatorname {div};\Omega ) \times {L^2}(\Omega )$ . Optimal order estimates are obtained for the approximation of *p* and the associated velocity field ${\mathbf {u}} = - (a\;{\mathbf {grad}}\;p + {\mathbf {b}}p)$ in ${L^2}(\Omega )$ and ${H^{ - s}}(\Omega )$, $0 \leqslant s \leqslant k + 1$, and, if $\Omega \subset {{\mathbf {R}}^2}$ for *p* in ${L^\infty }(\Omega )$.

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© Copyright 1985
American Mathematical Society