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The discontinuous Galerkin method with diffusion. (English) Zbl 0783.65078
Let $$\Omega\subset \mathbb{R}^ 2$$ be a bounded polygon and $$\alpha=(\alpha_ 1,\alpha_ 2)$$ a unit vector. The author considers the following class of constant-coefficient convection-diffusion equations: (1) $$u_ \alpha-\sigma_ 1u_{xx}-\sigma_ 2u_{yy}=f$$, where $$(x,y)\in \Omega$$, $$u_ \alpha=\alpha\cdot\bigtriangledown u$$ and $$\sigma_ 1$$ and $$\sigma_ 2$$ are nonnegative. Equation (1) may be hyperbolic, parabolic or elliptic depending upon the number of nonzero diffusion coefficients which appear. The following extension of the discontinuous Galerkin method is proposed to (1): $\begin{split} (u_ \alpha^ h- \sigma_ 1u_{xx}^ h- \sigma_ 2u_{yy}^ h,v^ h)- \int_{\Gamma_{in}(T)}[(u^ h)^ +-(u^ h)^ -]v^ h\alpha\cdot n+\\ \int_{\Gamma_{in}^*(T)}\{\sigma_ 1[(u_ x^ h)^ +-(u_ x^ h)^ -]n_ 1+\sigma_ 2[(u_ y^ h)^ +- (u_ y^ h)^ - ]n_ 2\}v^ h=(f,v^ h), \quad \text{for all}\quad v^ h\in P_ n(T).\end{split}\tag{2}$ Here $$\Gamma_{in}(\Omega)$$ is the “inflow” portion of $$\Gamma=\partial\Omega$$ defined by $$\alpha\cdot n<0$$, $$T$$ is a generic triangle of the triangulation, $$P_ n(T)$$ is the space of polynomials of degree $$\leq n$$, $$\Gamma_{in}^*(T)$$ denotes those sides of $$\Gamma_{in}(T)$$ which are not part of $$\Gamma_{in}(\Omega)$$; $$n_ 1$$, $$n_ 2$$ are $$x$$, $$y$$-components of the unit outer normal to $$T$$.
The main theorem of the paper provides error estimates for $$u^ h$$. The performed analysis is based on the nonalignment condition (3) $$| \alpha\cdot n| \geq c>0$$, which means that the triangle sides are to be bounded away from the characteristic direction ($$\Omega$$ is divided into a quasi-uniform mesh of triangles of side length $$h$$). Two specific examples (elliptic and parabolic equations) illustrate the used method.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35G15 Boundary value problems for linear higher-order PDEs
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##### References:
  Richard S. Falk and Gerard R. Richter, Analysis of a continuous finite element method for hyperbolic equations, SIAM J. Numer. Anal. 24 (1987), no. 2, 257 – 278. · Zbl 0619.65100 · doi:10.1137/0724021 · doi.org  T. J. R. Hughes and A. Brooks, A multidimensional upwind scheme with no crosswind diffusion, Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979) AMD, vol. 34, Amer. Soc. Mech. Engrs. (ASME), New York, 1979, pp. 19 – 35. · Zbl 0423.76067  Claes Johnson, Uno Nävert, and Juhani Pitkäranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 285 – 312. · Zbl 0526.76087 · doi:10.1016/0045-7825(84)90158-0 · doi.org  C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1 – 26. · Zbl 0618.65105  P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89 – 123. Publication No. 33. · Zbl 0341.65076  Todd E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1991), no. 1, 133 – 140. · Zbl 0729.65085 · doi:10.1137/0728006 · doi.org  W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479, 1973.  Gerard R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp. 50 (1988), no. 181, 75 – 88. · Zbl 0643.65059  Gerard R. Richter, A finite element method for time-dependent convection-diffusion equations, Math. Comp. 54 (1990), no. 189, 81 – 106. · Zbl 0738.65078  Gerard R. Richter, An explicit finite element method for convection-dominated steady state convection-diffusion equations, SIAM J. Numer. Anal. 28 (1991), no. 3, 744 – 759. · Zbl 0727.65079 · doi:10.1137/0728040 · doi.org  M. I. Višik and L. A. Ljusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Transl. (2) 20 (1962), 239 – 364.
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