A posteriori error bounds for discontinuous Galerkin methods for quasilinear parabolic problems

We derive a posteriori error bounds for a quasilinear parabolic problem, which is approximated by the $hp$-version interior penalty discontinuous Galerkin method (IPDG). The error is measured in the energy norm. The theory is developed for the semidiscrete case for simplicity, allowing to focus on the challenges of a posteriori error control of IPDG space-discretizations of strictly monotone quasilinear parabolic problems. The a posteriori bounds are derived using the elliptic reconstruction framework, utilizing available a posteriori error bounds for the corresponding steady-state elliptic problem.Comment: 8 pages, conference ENUMATH 200

Scalable Algorithms for the Solution of Naviers Equations of Elasticity

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Canonical eigenvalue distribution of multilevel block Toeplitz sequences with non-Hermitian symbols

Let $f:I_k \rightarrow {\mathcal M}_s$ be a bounded symbol with $I_k=(-\pi,\pi)^k$ and ${\mathcal M}_s$ be the linear space of the complex $s\times s$ matrices, $k,s\ge 1$. We consider the sequence of matrices $\{T_n(f)\}$, where $n=(n_1,\ldots,n_k)$, $n_j$ positive integers, $j=1\,\ldots,k$. Let $T_n(f)$ denote the multilevel block Toeplitz matrix of size $\widehat{n} \,s$, $\widehat{n}=\prod_{j=1}^k n_j$, constructed in the standard way by using the Fourier coefficients of the symbol $f$. If $f$ is Hermitian almost everywhere, then it is well known that $\{T_n(f)\}$ admits the canonical eigenvalue distribution with the eigenvalue symbol given exactly by $f$ that is $\{T_n(f)\}\sim_\lambda (f, I_k)$. When $s=1$, thanks to the work of Tilli, more about the spectrum is known, independently of the regularity of $f$ and relying only on the topological features of $R(f)$, $R(f)$ being the essential range of $f$. More precisely, if $R(f)$ has empty interior and does not disconnect the complex plane, then $\{T_n(f)\}\sim_\lambda (f, I_k)$. Here we generalize the latter result for the case where the role of $R(f)$ is played by $\bigcup_{j=1}^s R(\lambda_j(f))$, $\lambda_j(f)$, $j=1,\ldots,s$, being the eigenvalues of the matrix-valued symbol $f$. The result is extended to the algebra generated by Toeplitz sequences with bounded symbols. The theoretical findings are confirmed by numerical experiments, which illustrate their practical usefulness

Schur complement matrix and its (elementwise) approximation: a spectral analysis based on GLT sequences

Using the notion of the so-called spectral symbol in the Generalized Locally Toeplitz (GLT) setting, we derive the GLT symbol of the sequence of matrices {An} approximating the elasticity equations. Further, as the GLT class defines an algebra of matrix sequences and Schur complements are obtained via elementary algebraic operation on the blocks of An, we derive the symbol fS of the associated sequences of Schur complements {Sn} and that of its element-wise approximation