We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.

We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions $h$ and $\tau $. For this we show that the $L^2$-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

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The discrete minimal least-squares functional LS(f ; U ) is equivalent to the squared error | | u -U | | 2 in least-squares finite element methods and so leads to an embedded reliable and efficient a posteriori error control. This paper enfolds a spectral analysis to prove that this natural error estimator is asymptotically exact in the sense that the ratio LS(f ; U )/| | u -U | | 2 tends to one as the underlying mesh-size tends to zero for the Poisson model problem, the Helmholtz equation, the linear elasticity, and the time-harmonic Maxwell equations with all kinds of conforming discretizations. Some knowledge about the continuous and the discrete eigenspectrum allows for the computation of a guaranteed error bound C(\scrT )LS(f ; U ) with a reliability constant C(\scrT ) \leq 1/\alpha smaller than that from the coercivity constant \alpha . Numerical examples confirm the estimates and illustrate the performance of the novel guaranteed error bounds with improved efficiency.1. Introduction. The least-squares finite element method (LSFEM) approximates the exact solution u \in X to a partial differential equation by the discrete minimizer U \in X(\scrT ) of a least-squares functional LS(f ; \bullet ) over a discrete subspace X(\scrT ) \subset X. For the problems in this paper, namely the Poisson model problem, the Helmholtz equation, the linear elasticity, and the Maxwell equations, the functional LS(f ; \bullet ) is equivalent to the norm \| \bullet \| 2 X in X with equivalence constants \alpha and \beta . In particular, the discrete minimizer U \in X(\scrT ) satisfies \alpha \leq LS(f ; U )/\| u -U \| 2 X \leq \beta and the computable residual LS(f ; U ) leads to a guaranteed upper bound (GUB) \| u -U \| 2 X \leq \alpha - 1 LS(f ; U ) [3]. Table 1 displays computed upper and lower bounds of the quotient LS(f ; U )/\| u -U \| 2 X for a Poisson model problem and provides numerical evidence of asymptotic exactness of the least-squares residual LS(f ; U ). This experiment suggests that the GUB \alpha - 1 LS(f ; U ) is too pessimistic for \alpha - 1 = 1.442114.The first main result of this paper verifies that the ratio LS(f ; U )/\| u -U \| 2 X with the unique exact (resp., discrete) minimizer u (resp., U ) tends to one in the model problems from section 2 as the maximal mesh size \delta of the underlying regular \ast Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis"" under the project Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics"" (CA 151/22-1). The research of the second author was supported by the Studienstiftung des deutschen Volkes.

We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.