Semidiscrete schemes of the finite element method for degenerate hyperbolic equations (Q2496477)

Degenerate hyperbolic equations of the type \[ u^{\prime\prime}(t) -\sum_{i,j=1}^{n} {\partial \over \partial x_i} \left( a_{ij}(x,t){\partial u \over \partial x_j} \right) +a_0 u=f(x,t) \] (I think here is the misprint in the article in formula (5)) with zero initial conditions and zero Dirichlet boundary condition are investigated. The problem is studied in an \(n\)-dimensional space domain \(\Omega\) and with the following assumption: \[ a_{ij}=a_{ji},\;a_0\geq 0, f \in L_2(\Omega\times (0,T)) \] and \[ c_1 \rho^{2\alpha} | \xi| ^2 \leq \sum_{i,j=1}^{n} a_{ij}(x,t)\xi_i \xi_j \leq c_2\rho^{2\alpha} | \xi| ^2, \] where \( \rho=\rho(x)=\inf \{| x-y|, y \in \Gamma_0 \}\), \(\Gamma_0 \subset \Gamma=\partial\Omega\). This problem is degenerate for the coefficient \(\alpha >0\). The authors focus to the so called weak degeneration, \( (\alpha < {1\over2})\) . For this problem they propose a semi-discrete in space finite element numerical approximation. Under obvious regularity and a so-called condensation property of the triangulation of the form: \[ \gamma_1 h^{\nu} \leq h_K \leq \gamma_2 h \rho_K^{1-{1 \over \nu}} \] for all triangles \(K\) where \(h=\max\{ h_K, K \in T_h \},\;\nu \geq 1\), the authors investigate a convergence analysis of the numerical solution and prove some error estimates in appropriate functional spaces under some regularity assumptions.