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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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