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A wavelet-based stabilization of the mixed finite element method with Lagrange multipliers. (English) Zbl 1101.65106
The authors consider the approximation of a second-order linear elliptic boundary value problem with a Neumann boundary condition. The problem is a model for many physical phenomena, one example being steady heat conduction in an anisotropic, non-homogeneous medium. The problem is approached through a mixed formulation, and the Neumann condition is imposed weakly through the use of a Lagrange multiplier.
I. Babuska and G. N. Gatica [Numer. Methods Partial Differ. Equations 19, 192–210 (2003; Zbl 1021.65056)] showed that the mixed problem under consideration is well-posed, and the associated finite element approximation is also well-posed with the choice of Raviart-Thomas elements of order zero for the flux variable, piecewise constants for the original variable, and continuous piecewise polynomials of degree at most one for the boundary Lagrange multiplier. The last is done on an independent partition of the domain, and stability holds only under suitable assumptions on the two mesh sizes associated with the two partitions.
The main goal of the work under review is that of removing this difficulty by introducing a stabilization. The original formulation is perturbed through the introduction of a residual equation for the Lagrange multiplier, expressed in terms of the inner product corresponding to the trace space on the boundary. This inner product is defined through the use of a biorthogonal wavelet basis. The addition also of residuals corresponding to the original partial differential equation and the divergence of the flux variable leads to a new mixed formulation which is in the standard format. Well-posedness of the continuous problem is established.
A finite element scheme corresponding to the mixed formulation is introduced and analyzed next. The scheme is based on a regular triangulation of the domain and a truncated form of the trace inner product involving a finite number of terms of the biorthogonal basis. By application once more of the Babuska-Brezzi theory well-posedness is established, as is a rate of convergence of \(O(h)\) for suitable choices of low-order elements.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65T60 Numerical methods for wavelets
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[1] Babuška, I.; Gatica, G.N., On the mixed finite element method with Lagrange multipliers, Numerical methods for partial differential equations, 19, 2, 192-210, (2003) · Zbl 1021.65056
[2] Bertoluzza, S., Wavelet stabilization of the Lagrange multiplier method, Numerische Mathematik, 86, 1-28, (2000) · Zbl 0966.65075
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[6] T.P. Barrios, G.N. Gatica, F. Paiva, A-priori and a-posteriori error analysis of a wavelet-based stabilization for the mixed finite element method (in preparation) · Zbl 1119.65107
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