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Applications of discrete maximal \(L_p\) regularity for finite element operators. (English) Zbl 1132.65093

This paper concerns the initial value problem of the form
\[ u'(t)-A_qu(t)=f(t),\;\;t\geq0\tag{1} \]
\[ u(0)=u_0,\tag{2} \]
where the operator \(A_q\) is defined by means of the standard elliptic sesquilinear form \(a(u,v)\) defined on the space \(H^1_0(\Omega)\) (or \(H^1(\Omega)\)), involving the function \(u\) and its first and second space derivatives. Under suitable assumptions, the operator \(A_q\) is a generator of a semigroup acting on the space \(L_q(\Omega)\). The aim of the paper is to give error estimates for \(u\) by following a finite element version of the problem (1), (2) \[ u_h'(t)-A_hu_h(t)=f_h,\;\;t\in {\mathbf I},\tag{3} \]
\[ u_h(0)=0,\tag{4} \]
(\({\mathbf I}\) is a bounded interval) in terms of the space \(L_p({\mathbb R}^{+},L_q(\Omega))\), \(1<p,q<\infty\). The basic tool used in this work is the maximal \(L_p\) regularity of the problem (3), (4) that is its property expressed in form of the following inequality (of the stability character)
\[ \| u'_h\| _{L_p(I;L_q(\Omega))}+\| A_h\| _{L_p(I;L_q(\Omega))} \leq C_h\| f_h\| _{L_p(I;L_q(\Omega))}\tag{5} \]
with constants \(C_h\) mostly assumed to be uniformly bounded. Theorem 3.5 of the paper states conditions under which (5) can be satisfied while Theorem 4.9 gives the convergence estimate of the standard form – the main result in the linear case
\[ \| u_h-u\| _{L_p({\mathbb R}^+;L_q(\Omega))}\leq Ch^2\left( \| f\| _ {L_p({\mathbb R}^+;L_q(\Omega))}+\| u_0\| _{(L_q(\Omega),D(A))_{1-{1\over p},p}}\;\right).\tag{6} \]
The above result can be generalized under additional assumptions about \(f\) (Lipschitz condition, etc.) for the nonlinear case where \(f\) (in finite element formulation) depends on \(u_h\) and its first space and time derivatives. The appendix to this paper is devoted to properties of the so called Ritz projections, extensively used in the proofs.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
34G25 Evolution inclusions
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K90 Abstract parabolic equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
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