Singular finite element methods.

*(English)*Zbl 0727.73070
Finite elements. Theory and application, Proc. ICASE Workshop, Hampton/VA 1986, 50-66 (1988).

[For the entire collection see Zbl 0657.00012.]

The key property of elliptic systems is that their solution tends to be as smooth as the data and other factors permit. This condition strikingly constrasts with hyperbolic systems, in which singular behavior (e.g., shocks) can arise even if all inputs are smooth.

To illustrate this, consider the following model problem defined in a bounded region \(\Omega\) of \({\mathbb{R}}^ n:\) div[a grad \(\phi\) ]\(=f\) in \(\Omega\), \(B[\phi]=g\) on \(\partial \Omega\). The boundary operator B has the form \(B[\phi]=\alpha (\partial \phi /\partial v)+\beta \phi\), where v is the outer normal to \(\partial \Omega\). Thus the inputs to this system are the coefficients a, \(\alpha\), \(\beta\) ; the data f, g; and the region \(\Omega\). If all these are smooth, then the same is true of the solution \(\phi\).

This means that singular behavior can arise only in the cases where the inputs are irregular. Four examples of technical importance are considered: 1. Irregular boundaries. 2. Discontinuous boundary operators. 3. Discontinuous coefficients. 4. Nonsmooth data.

Sections 3.2 and 3.3 are devoted to the types of singular behavior that can arise in elliptic systems. Section 3.3 considers nonlinear effects, and this material is apparently new. The final section concentrates on practical issues associated with singular elements along with selected numerical results.

The key property of elliptic systems is that their solution tends to be as smooth as the data and other factors permit. This condition strikingly constrasts with hyperbolic systems, in which singular behavior (e.g., shocks) can arise even if all inputs are smooth.

To illustrate this, consider the following model problem defined in a bounded region \(\Omega\) of \({\mathbb{R}}^ n:\) div[a grad \(\phi\) ]\(=f\) in \(\Omega\), \(B[\phi]=g\) on \(\partial \Omega\). The boundary operator B has the form \(B[\phi]=\alpha (\partial \phi /\partial v)+\beta \phi\), where v is the outer normal to \(\partial \Omega\). Thus the inputs to this system are the coefficients a, \(\alpha\), \(\beta\) ; the data f, g; and the region \(\Omega\). If all these are smooth, then the same is true of the solution \(\phi\).

This means that singular behavior can arise only in the cases where the inputs are irregular. Four examples of technical importance are considered: 1. Irregular boundaries. 2. Discontinuous boundary operators. 3. Discontinuous coefficients. 4. Nonsmooth data.

Sections 3.2 and 3.3 are devoted to the types of singular behavior that can arise in elliptic systems. Section 3.3 considers nonlinear effects, and this material is apparently new. The final section concentrates on practical issues associated with singular elements along with selected numerical results.