A remark on split least-squares mixed element procedures for pseudo-parabolic equations
The author applies a split least-squares Galerkin procedure, previously constructed for stationary diffusion reaction problems and parabolic problems, and give two novel split least-squares mixed element procedures to solve pseudo-parabolic equations over \((\Omega\times J)\), where \(\Omega\) is a bounded polygonal domain in \(\mathbb{R}^d\) (\(d=2,3)\) and \(J=(0,T]\) is the time interval. By a proper selection of the least-square functional property, the resulting least-squares mixed element procedure can be split into two symmetric positive definite sub-schemes, one of which is for the primary unknown \(u\) and the other is for the introduced unknown flux variable \(\sigma\). In the first procedure, the sub-procedure for \(u\) is the same as the Crank--Nicolson standard Galerkin finite element procedure with first-order approximation in time increment, while in the second procedure one has second-order approximation in time increment. The convergence analysis shows that both methods lead to optimal error estimates for \(u\) in the \(H^1(\Omega)\) and \(L^2(\Omega)\) norms and optimal error estimate for \(\sigma\) in the \((\text{div};\Omega)\) norm. A numerical example shows good agreement with the theoretical analysis.
scientific article; zbMATH DE number 5843693
scientific article