Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data
The following parabolic problem in a Hilbert space \((X,\|\|)\) is considered \((1)\quad u'(t)=-Au(t)+f(u(t)),\) \(u(0)=v\in X\) where \(A\) is linear, selfadjoint and positive definite, \(f: V\to X\) is nonlinear, satisfying the conditions \(f(0)=0\), and: \[ (2)\quad \exists a<\alpha <2,\quad \forall R>0,\quad \exists C(R)>0,\quad \forall \epsilon >0,\quad \forall u,v\quad J(\epsilon,u),J(\epsilon,v)<R,\quad \| f(u)-f(v)\| \leq C(R)\epsilon^{\alpha -}J(\epsilon,u-v) \] with \(J(\epsilon,u)=Max(\| u\|,\epsilon \| v\|_ V)\). The space \((V,\|\|_ V)\) is the domain of definition of the bilinear form \(a\), defining in the standard way the ``generalized operator \(A\)''. The first result (Theorem 1) is an existence and uniqueness theorem for the solution of (1) with the only assumption on \(v: v\in X\). Also an estimate for the solution \(u(t)\) of the following form \(\| u(t)\| +t^{1/2}\| u(t)\|_ V\leq C(v)\) \(t\in (0,t_ 0]\) is derived. Applying standard Galerkin techniques on finite element type subspaces of \(X\), a semidiscrete (dependent on \(t\)) approximation \(u_ h(t)\) is constructed for (1). Then, the author introduces certain intermediate spaces Y, \(V\subset Y\subset X\), depending on some parameter \(\beta >0\), and strengthens the condition (2) for f, making it dependent on Y. The second result (Theorem 2) are following estimates for the Galerkin error operator \(e_ h(t):\| e_ h(t)\| \leq Ch^{\mu}t^{-\mu /2}\) and \(\| e_ h(t)\|_ Y\leq Ch^{\mu}t^{-(\mu +\beta)/2}\) (\(\mu\geq 0\)). Two examples are examined.
scientific article; zbMATH DE number 4058789
scientific article