The effect of numerical integration in finite element methods for nonlinear Sobolev equations (Q2721779)

This paper deals with the numerical approximation of solutions of nonlinear initial boundary value Sobolev (evolution) equations. Their weak formulation looks for a solution \(u\) defined on the time interval and with values in \(H^1_0(\Omega)\), where \(\Omega \) is a polygonal domain in Euclidean space. By means of some quadrature rule on \(\Omega \), one can substitute the integrals in the weak formulation and look for a solution \(U\) of this discretized weak formulation, whose values are in a finite element subspace of functions in \(H^1_0(\Omega)\), which are continuous on the closure of \(\Omega \). NEWLINENEWLINENEWLINEA typical result proved in the paper is that if the quadrature rule is exact in \(P_{2k}\), i.e. the space of polynomials of degree smaller or equal than \(2k\), and the set of points in \(\Omega \) used in the rule contains a unisolvent subset for \(P_k(\Omega)\), and if \(u\) and \(u_t\) are in \(L_{\infty }(W^{k+1,\infty }(\Omega))\), then for the finite element method mesh size \(h\) sufficiently small, there exists a constant \(C=C(u)\), such that NEWLINE\[NEWLINE\|u-U\|_{L_{\infty }(L_2)}+\|u_t-U_t\|_{L_{\infty }(L_2)}\leq Ch^{k+1},NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\|u-U\|_{L_{\infty }(H^1)}+\|u_t-U_t\|_{L_{\infty }(H^1)}\leq Ch^k.NEWLINE\]