Constantin Bacuta

A Unified Approach for Uzawa Algorithms

We present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite‐ or infinite‐dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition.

Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT Let 𝒯 k be a sequence of triangulations of a polyhedron Ω ⊂ ℝ n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u k ∈ S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.

Regularity estimates for elliptic boundary value problems in Besov spaces

We consider the Dirichlet problem for Poissons equation on a nonconvex plane polygonal domain Ω. New regularity estimates for its solution in terms of Besov and Sobolev norms of fractional order are proved. The analysis is based on new interpolation results and multilevel representations of norms on Sobolev and Besov spaces. The results can be extended to a large class of elliptic boundary value problems. Some new sharp finite element error estimates are deduced.

Improving the rate of convergence of `high order finite elements' on polyhedra II: Mesh refinements and interpolation

We construct a sequence of meshes 𝒯 k ′ that provides quasi-optimal rates of convergence for the solution of the Poisson equation on a bounded polyhedral domain with right-hand side in H m−1, m ≥ 2. More precisely, let Ω ⊂ ℝ3 be a bounded polyhedral domain and let u ∈ H 1(Ω) be the solution of the Poisson problem − Δ u = f ∈ H m−1(Ω), m ≥ 2, u = 0 on ∂ Ω. Also, let S k be the finite element space of continuous, piecewise polynomials of degree m ≥ 2 on 𝒯 k ′ and let u k ∈ S k be the finite element approximation of u, then ‖u − u k ‖ H 1(Ω) ≤ C dim(S k )−m/3 ‖f‖ H m−1(Ω), with C independent of k and f. Our method relies on the a priori estimate ‖u‖𝒟 ≤ C ‖f‖ H m−1(Ω) in certain anisotropic weighted Sobolev spaces , with a > 0 small, determined only by Ω. The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in 𝒯 k ′ will be divided into two segments AC and CB in 𝒯 k+1′ as follows: |AC| = |CB| if A and B are equally singular and |AC| = κ |AB| if A is more singular than B. We can choose κ ≤ 2−m/a . This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct 𝒯 k ′.

Shift Theorems for the Biharmonic Dirichlet Problem

We consider the biharmonic Dirichlet problem on a polygonal domain. Regularity estimates in terms of Sobolev norms of fractional order are proved. The analysis is based on new interpolation results which generalizes Kellogg’s method for solving subspace interpolation problems. The Fourier transform and the construction of extension operators to Sobolev spaces on R 2 are used in the proof of the interpolation theorem.

Regularity estimates for elliptic boundary value problems with smooth data on polygonal domains

We consider the model Dirichlet problem for Poissons equation on a plane polygonal convex domain Ω with data ƒ in a space smoother than L 2. The regularity and the critical case of the problem depend on the measure of the maximum angle of the domain. Interpolation theory and multilevel theory are used to obtain estimates for the critical case. As a consequence, sharp error estimates for the corresponding discrete problem are proved. Some classical shift estimates are also proved using the powerful tools of interpolation theory and mutilevel approximation theory. The results can be extended to a large class of elliptic boundary value problems.

New interpolation results and applications to finite element methods for elliptic boundary value problems

Abstract We consider the the interpolation problem between and , where Ω is a polygonal domain in and is the subspace of functions in H 1(Ω) which vanish on the Dirichlet part (∂Ω) D of the boundary of Ω. The main result is that the interpolation spaces and coincide. An application of this result to a nonconforming finite element problem is presented.

Using finite element tools in proving shift theorems for elliptic boundary value problems

We consider the Laplace equation under mixed boundary conditions on a polygonal domain Ω. Regularity estimates in terms of Sobolev norms of fractional order for this type of problem are proved. The analysis is based on new interpolation results and multilevel representation of norms on the Sobolev spaces Hα(Ω). The Fourier transform and the construction of extension operators to Sobolev spaces on ℝ2 are avoided in the proofs of the interpolation theorems. Copyright

Residual reduction algorithms for nonsymmetric saddle point problems

In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state Navier-Stokes equations.

Cascadic multilevel algorithms for symmetric saddle point systems

Abstract In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems . On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non-increasing number of iterations on each level. Even though numerical results supporting the efficiency of the algorithms are presented for the Stokes system, the algorithms can be applied to a large class of boundary value problems, including first order systems that can be reformulated at the continuous level as symmetric saddle point problems, such as the Maxwell equations.