اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA new framework for the stability analysis of perturbed saddle-point problems and applications
102
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we prove a new abstract stability result for perturbed saddle-point problems based on a norm fitting technique. We derive the stability condition according to Babuv{s}kas theory from a small inf-sup condition, similar to the famous Ladyzhenskaya-Babuv{s}ka-Brezzi (LBB) condition, and the other standard assumptions in Brezzis theory, in a combined abstract norm. The construction suggests to form the latter from individual {it fitted} norms that are composed from proper seminorms. This abstract framework not only allows for simpler (shorter) proofs of many stability results but also guides the design of parameter-robust norm-equivalent preconditioners. These benefits are demonstrated on mixed variational formulations of generalized Poisson, Stokes, vector Laplace and Biots equations.
قيم البحث
اقرأ أيضاً
In this paper, two types of Schur complement based preconditioners are studied for twofold and block tridiagonal saddle point problems. One is based on the nested (or recursive) Schur complement, the other is based on an additive type Schur complemen
t after permuting the original saddle point systems. We discuss different preconditioners incorporating the exact Schur complements. It is shown that some of them will lead to positive stable preconditioned systems. Our theoretical analysis is instructive for devising various exact and inexact preconditioners, as well as iterative solvers for many twofold and block tridiagonal saddle point problems.
We introduce an adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. The algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are
sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom.
In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using integral-value degrees of freedom for solving elliptic interface problems. We show that those IFE methods can only achieve suboptimal c
onvergence rates (i.e., $O(h^{1/2})$ in the $H^1$ norm and $O(h)$ in the $L^2$ norm) if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The unisolvence of IFE basis functions is proved on arbitrary triangles. Furthermore, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-$Q_1$ IFE spaces for interface problems with variable coefficients via a unified approach different from multipoint Taylor expansions. Finally, optimal error estimates in both $H^1$- and $L^2$- norms are proved and confirmed with numerical experiments.
For the nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) parts, the modified generalized shift-splitting (MGSSP) preconditioner as well as the MGSSP iteration method are derived in this paper, which generalize the MSSP pre
conditioner and the MSSP iteration method newly developed by Huang and Su (J. Comput. Appl. Math. 2017), respectively. The convergent and semi-convergent analysis of the MGSSP iteration method are presented, and we prove that this method is unconditionally convergent and semi-convergent. In addition, some spectral properties of the preconditioned matrix are carefully analyzed. Numerical results demonstrate the robustness and effectiveness of the MGSSP preconditioner and the MGSSP iteration method, and also illustrate that the MGSSP iteration method outperforms the GSS and GMSS iteration methods, and the MGSSP preconditioner is superior to the shift-splitting (SS), generalized SS (GSS), modified SS (MSS) and generalized MSS (GMSS) preconditioners for the GMRES method for solving the nonsymmetric saddle point problems.
In this paper, by employing the asymptotic method, we prove the existence and uniqueness of a smoothing solution for a singularly perturbed Partial Differential Equation (PDE) with a small parameter. As a by-product, we obtain a reduced PDE model wit
h vanished high order derivative terms, which is close to the original PDE model in any order of this small parameter in the whole domain except a negligible transition layer. Based on this reduced forward model, we propose an efficient two step regularization algorithm for solving inverse source problems governed by the original PDE. Convergence rate results are studied for the proposed regularization algorithm, which shows that this simplification will not (asymptotically) decrease the accuracy of the inversion result when the measurement data contains noise. Numerical examples for both forward and inverse problems are given to show the efficiency of the proposed numerical approach.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
