Necessary \(L^ 2\)-uniform convergence conditions for difference schemes for two-dimensional convection-diffusion problems
On the square \((0, 1)^ 2\) the authors consider the following singularly perturbed elliptic problem: \(\varepsilon \Delta u+ pu_ x+ qu_ y= f\) with homogeneous Dirichlet boundary condition, where \(0< \varepsilon<<1\), \(f\in L^ 2\), \(p\) and \(q\) are positive constants. A linear difference scheme based on nine-points stencil with coefficients \(a_{\nu,\mu}\) depending only on \(\rho\) \((= h/\varepsilon)\) is associated with the above problem. (Here \(h\) is a mesh spacing.) The main result is as follows: uniform (i.e. independent of \(\varepsilon\)) convergence of order \(h^{{1\over 2}}\) of the discrete solution in some (local) discrete \(L^ 2\) norm implies two conditions on the coefficients. The first one is like this: \[ \Biggl(\sum^ 1_{\mu= -1} a_{- 1,\mu}\Biggr) e^{p\rho}+ \sum^ 1_{\mu= -1} a_{0, \mu}+ \Biggl( \sum^ 1_{\mu= -1} a_{1, \mu}\Biggr) e^{-p\rho}= 0. \] The second one is similar. Next, for a wide class of basis functions spanning the finite element trial space the authors prove the equivalence of discrete \(L^ 2\) and continuous \(L^ 2\) norms. It enables them to obtain a ``continuous'' counterpart of the previous result, which is applied to study a streamline-diffusion finite element method.
scientific article; zbMATH DE number 737491
scientific article