On a mixed finite element method for the \(p\)-Laplacian (Q2730914)

For \(1 < p < \infty \), the quasilinear equation NEWLINE\[NEWLINE-\nabla \cdot (|\nabla u |^{p-2} \nabla u) = f \quad \text{in }\Omega, \qquad u = 0 \quad \text{on } \GammaNEWLINE\]NEWLINE is known as the \(p-\)Laplace equation for real \(u\) defined on the two or three-dimensional open set \( \Omega \) which has a Lipschitz boundary \(\Gamma \). (Here \(\nabla u\) denotes the gradient of \(u\) and \(|\cdot|\) the Euclidean norm.) Such an equation is present in many mathematical models, among them: nonlinear diffusion and filtration, power-law materials and viscoelastic materials. NEWLINENEWLINENEWLINEThis article studies a mixed finite element method for the \(p-\)Laplacian, improving estimate results previously obtained by different authors. In short, they employ Raviart-Thomas elements to get approximations to the solution of the so-called mixed formulation, which consists on introducing NEWLINE\[NEWLINE \sigma = |\nabla u |^{p-2} \nabla u NEWLINE\]NEWLINE or equivalently, for \(p, q\) conjugate gradients, NEWLINE\[NEWLINE \nabla u = |\sigma |^{q-2} \sigma .NEWLINE\]NEWLINE Such a procedure takes the \(p-\)Laplace equation to the first order system NEWLINE\[NEWLINE|\sigma |^{q-2} \sigma = \nabla u \quad \text{in } \Omega,\qquad \nabla \cdot \sigma + f = 0\quad \text{in } \Omega, \qquad u = 0 \quad \text{in } \Gamma ,NEWLINE\]NEWLINE whose variational formulation is NEWLINE\[NEWLINE \begin{aligned} \int _{\Omega} A(\sigma)\cdot \tau dx + \int_{\Omega} \nabla \cdot \tau u dx& = 0 \quad\forall \tau \in X \\ \int _{\Omega} \nabla \cdot \sigma v dx + \int _{\Omega} f v dx &=0 \quad \forall v \in L^p(\Omega) \end{aligned}\tag{1} NEWLINE\]NEWLINE It is assumed that \(f \in L^q(\Omega)\), while \(A(\sigma)\) denotes \(|\sigma |^{q-2} \sigma \), \(N \) is the space dimension (2 or 3) and NEWLINE\[NEWLINE X = \{\tau \in (L^q(\Omega))^N;\;\nabla \cdot \tau \in L^q(\Omega)\} . NEWLINE\]NEWLINE Within this framework, the following result is reached: NEWLINENEWLINENEWLINETheorem. Let the solution \(( \sigma , u) \) of (1) belong to \( (W^{1,q}(\Omega))^N \times W^{1,p}(\Omega)\) and denote by \((\sigma_h ,u_h)\) its approximation with the lowest degree finite-element of Raviart-Thomas. We guarantee the existence of a constant \(C\) which independs of the discretization parameter \(h\) and for which the estimates NEWLINE\[NEWLINE \|\sigma - \sigma _h\|_{q} \leq \begin{cases} C h^{q/2} \\ C h^{2/q}\end{cases}, \qquad \|u - u _h\|_{p} \leq \begin{cases} C h^{q-1} \\ C h \end{cases} NEWLINE\]NEWLINE hold, the first and third ones for \(2\leq p < \infty\), and the other two for \(1<p<2\).