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Superconvergence phenomenon in the finite element method arising from averaging gradients. (English) Zbl 0575.65104
The elliptic boundary value problem -div(A grad u)$$=f$$ in $$D\subset {\mathbb{R}}^ 2$$, $$u=0$$ on $$\partial D$$, is solved by the finite element method. (A is an uniformly positive definite $$2\times 2$$ matrix of smooth functions). The usual spaces of linear spline functions over uniform triangular meshes are used. The averaged gradient of a spline function is a piecewise linear continuous vector field, its value at any nodal point is an average of values of the gradient of the spline in the triangles incident with this nodal point. It is shown that the gradient of a smooth function can be approximated by an averaged gradient with a rate of $$O(h^ 2)$$ in $$L^ 2$$. This is used to prove local superconvergence for the averaged gradient of the Ritz-Galerkin approximation to the exact gradient of the solution u $$(O(h^{3/2})$$ in $$L^ 2$$ if u is sufficiently smooth, $$O(h^ 2)$$ if D is a parallelogram). It is remarked that the same results can be obtained for boundary conditions of Newton type. The given numerical examples confirm the proved theoretical accuracy.
Reviewer: J.Weisel

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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##### References:
 [1] de Boor, C., Swartz, B.: Collocation at Gaussian points. SIAM J. Numer. Anal.10, 582–606 (1973) · Zbl 0232.65065 [2] Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 112–124 (1970) · Zbl 0201.07803 [3] Bramble, J.H., Schatz, A.H.: Estimates for spline projection. RAIRO Anal. Numér.10, 5–37 (1976) · Zbl 0343.65045 [4] Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comput.31, 94–111 (1977) · Zbl 0353.65064 [5] Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam-New York-Oxford: North-Holland 1978 · Zbl 0383.65058 [6] Giarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems. Numer. Math.9, 394–430 (1967) · Zbl 0155.20403 [7] Dautov, R.Z.: Superconvergence of finite-element method schemes with numerical integration for quasilinear fourth-order elliptic equations. Differential Equations18, 818–824 (1982) · Zbl 0529.65067 [8] Dautov, R.Z., Lapin, A.V.: Difference schemes of an arbitrary order of accuracy for quasilinear elliptic equations (Russian). Izv. Vysš. Učebn. Zaved. Matematika209, 24–37 (1979) · Zbl 0477.65071 [9] Dautov, R.Z., Lapin, A.V.: Investigation of the convergence, in mesh norms, of finite-element-method schemes with numerical integration for fourth-order elliptic equations. Differential Equations17, 807–817 (1981) · Zbl 0492.65061 [10] Descloux, J.: Interior regularity and local convergence of Galerkin finite element approximations for elliptic equations. Topics in Numerical Analysis II. New York: Academic Press, pp. 27–41, 1975 · Zbl 0343.65049 [11] Douglas, J.J.: A superconvergence result for the approximate solution of the heat equation by a collocation method. Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. (Proc. Sympos. Univ. of Maryland, 1972, Aziz, A.K. (ed.)). New York-London: Academic Press, pp. 475–490, 1972 [12] Douglas, J.J., Dupont, T.: Superconvergence for Galerkin methods for the two point boundary problem via local projections. Numer. Math.21, 270–278 (1973) · Zbl 0281.65046 [13] Douglas, J.J., Dupont, T.: Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problems. In: Topics in Numerical Analysis. Miller, J.J.H. (ed.), pp. 89–92. London: Academic Press 1973 [14] Douglas, J.J., Dupont, T.: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math.22, 99–109 (1974) · Zbl 0331.65051 [15] Douglas, J.J., Dupont, T., Wheeler, M.F.: Some super-convergence results for anH 1-Galerkin procedure for the heat equation. Computing Methods in Engineering, Part 1 (Proc. Sympos., Versailles, 1973). Berlin-Heidelberg-New York: Springer 1974, pp. 288–311 [16] Douglas, J.J., Dupont, T., Wheeler, M.F.: AnL estimate and a superconvergence result for a Galerkin method for elliptic equations based on tensor products of piecewise polynomials. RAIRO Anal. Numér.8, 61–66 (1974) · Zbl 0315.65062 [17] Houstis, E.N.: Application of method of collocation on lines for solving nonlinear hyperbolic problems. Math. Comput.31, 443–456 (1977) · Zbl 0358.65099 [18] Johson, C., Pitkäranta, J.: Analysis of some mixed finite element methods related to reduced integration. Math. Comput.33, 375–400 (1982) · Zbl 0482.65058 [19] Lapin, A.V.: Schemes of the finite element methods for some classes of variational inequalities, error estimates, algorithms (Russian). (Proc. of the fifth conference on variational-difference methods in mathematical physics, Moscow. 1983) (To appear in 1984) [20] Lesaint, P., Zlámal, M.: Superconvergence of the gradient of finite element solutions. RAIRO Anal. Numér.13, 139–166 (1979) · Zbl 0412.65051 [21] Lindberg, B.: Error estimation and iterative improvement for discretization algorithms. BIT20, 486–500 (1980) · Zbl 0459.65036 [22] Long, M.J., Morton, K.W.: The use of divided differences in finite element calculations. J. Inst. Math. Appl.19, 307–323 (1977) · Zbl 0354.65055 [23] Nečas, J.: Les méthodes directes en théorie des equations elliptiques. Masson, Paris, 1967 [24] Neittaanmäki, P., Saranen, J.: A modified least squares FE-method for ideal fluid flow problems. J. Comput. Appl. Math.8, 165–169 (1982) · Zbl 0488.76004 [25] Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz-Galerkin methods. Math. Comput.28, 937–958 (1974) · Zbl 0298.65071 [26] Oganesjan, L.A., Ruhovec, L.A.: An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary (Russian). Ž. Vyčisl. Mat. i Mat. Fiz.9, 1102–1120 (1969) [27] Oganesjan, L.A., Ruhovec, L.A.: Variational-difference methods for the solution of elliptic equations (Russian). Izd. Akad. Nauk Armjanskoi SSR, Jerevan, 1979 [28] Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput.31, 414–442 (1977) · Zbl 0364.65083 [29] Strang, G., Fix, G.: An analysis of the finite element method. Englewood Cliffs-New Jersey: Prentice Hall 1973 · Zbl 0356.65096 [30] Thomée, V.: Spline approximation and difference schemes for the heat equation. Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Proc. Sympos., Univ. of Maryland, 1972, A.K. Aziz (ed.). New York-London: Academic Press, 1972, pp. 711–746 [31] Thomée, V.: Some error estimates in Galerkin methods for parabolic equations. Mathematical Aspects of Finite Element Methods (Proc. of the the conference, Rome, 1975). Berlin-Heidelberg-New York: Springer 1977, pp. 353–362 [32] Thomée, V.: High order local approximations to derivatives in the finite element method. Math. Comput.31, 652–660 (1977) · Zbl 0367.65055 [33] Thomée, V., Wendroff, B.: Convergence estimates for Galerkin method for variable coefficient initial value problems. SIAM J. Numer. Anal.11, 1059–1068 (1974) · Zbl 0292.65052 [34] Vacek, J.: Dual variational principles for an elliptic partial differential equation. Apl. Mat.21, 5–27 (1976) · Zbl 0345.35035 [35] Volkov, E.A.: A rapidly converging quadratures for solving Laplace’s equation on polygons. Soviet Math.19, 154–157 (1978) · Zbl 0409.65049 [36] Wendland, W.L., Stephan, E., Hsiao, G.C.: On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci.1, 265–321 (1979) · Zbl 0461.65082 [37] Zienkiewicz, O.C., Cheung, Y.K.: The finite element method in structural and continuum mechanics. London: McGraw Hill 1967 · Zbl 0189.24902 [38] Zlámal, M.: Some superconvergence results in the finite element method. Mathematical Aspects of Finite Element Methods (Proc. of the conference, Rome, 1975). Berlin-Heidelberg-New York: Springer 1977, pp. 353–362 [39] Zlámal, M.: Superconvergence and reduced integration in the finite element method. Math. Comput.32, 663–685 (1978) · Zbl 0448.65068
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