A charcoal kiln made of Cinder-Concrete blocks by A. Richard Olson, Henry W. Hicock

By A. Richard Olson, Henry W. Hicock

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38) is positive definite. Then, show that if the bilinear form a(·, ·) is symmetric, so is the matrix Ah . 10. e. to the nonsingularity of the matrix Ah . e. to the nonsingularity of ATh . Finally, show that for a square matrix these two conditions are equivalent. 11. 60) is not fulfilled, the solution of problem (P3h ) is not unique. Hence, supposing that inf sup qh ∈Qh wh ∈Xh b(wh , qh ) =0 wh X qh Q show that there exists (at least one) q∗h ∈ Qh such that b(xh , q∗h ) = 0 for any xh ∈ Xh ; then, conclude that if (xh , ph ) ∈ Xh × Qh is solution to (P3h ), the same happens for (xh , ph + q∗h ).

4 On the Numerical Approximation of Variational Problems 25 spaces is required. 34) (also known as C´ea’s Lemma), in order for the method to converge it will be sufficient to require that, for h → 0, the space Vh tends to “fill” the entire space V . Precisely, lim inf v − vh h→0 vh ∈Vh V =0 ∀v ∈ V. 35). In this way, by taking a sufficiently small h, it is possible to approximate u by uh as accurately as desired. The actual convergence rate will depend on the specific choice of the subspace Vh . In the finite element case in which the latter is made of piecewise polynomials of degree r ≥ 1, u − uh V will tend to zero as O(hr ), see Sect.

2 Algebraic Form of (P1h ) The discrete variational problem (P1h ) is equivalent to the solution of a linear system Nh a basis for the finite-dimensional of equations. Indeed, if we denote by {ϕ j } j=1 space Vh , then every vh ∈ Vh has a unique representation Nh ( j) ∑ vh vh = (1) ϕ j, j=1 N (Nh ) T with v = (vh , . . , vh ) ∈ RNh . ( j) h uh ϕ j , and denoting by uh the vector having as components By setting uh = ∑ j=1 ( j) the unknown coefficients uh , (P1h ) is equivalent to: find uh ∈ RNh such that Nh ( j) ∑ a(ϕ j , ϕ i )uh = f (ϕ i ) ∀ i = 1, .

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