A characteristic mixed method with dynamic finite-element space for convection-dominated diffusion problems
The method of characteristics is combined with the dynamic mixed finite element method to solve the model problem \(c(x)\partial u/\partial t+\sum^ 2_{i=1}b_ i(x)\partial u/\partial x_ i-\nabla_ 0(d(x)\nabla u)=f(x,t)\), \((x,t)\in\Omega\times J\), \(u(x,t)=0\), \((x,t)\in\Gamma\times J\), \(u(x,0)=g(x)\), \(x\in\Omega\), where \(\Omega\) is a polygonal domain in \(\mathbb{R}^ 2\) with boundary \(\Gamma\) and \(J=[0,T]\). Discretization of the transport terms along the characteristic direction is used and the mixed method for approximation of the solution and its gradient is applied. Different grids and different basis functions are implemented at different time levels in order to accurately resolve time- changing critical features. Convergence analysis and error estimates are established and some comments on the efficiency and capability of the method are made, as well.
scientific article; zbMATH DE number 85940
scientific article