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FEM:010117Two P1 elements for -u''+u=1 with Robin: free A [[13/3,-23/12],[-23/12,25/6]], u=(0,63/109,114/109), reaction -148/109

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Fem Robin Reaction Diffusion AssemblyFem Constrained Row Discrete Reaction

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010118One-mode Ritz for -u''=x: a=1/4, uh=x(1-x)/4, min J=-1/96, energy-error squared 1/720

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Fem One Mode Ritz PoissonFem Energy Error Galerkin Orthogonality

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010119Symmetric Nitsche on one P1 interval: stiffness 3, load 6, uh=2(1-x), coercive here iff gamma>1

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Fem Symmetric Nitsche DirichletFem Nitsche One Mode Coercivity

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010220P1 triangle (0,0),(2,0),(0,1) with k=3,q=6: K=[[15/4,-3/4,-3],[-3/4,3/4,0],[-3,0,3]], energy 39

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Fem P1 Triangle StiffnessFem Affine Gradient Energy

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010221Quadratic [0,1] Gauss: K exact at 2 pts, M2 singular on (2,-1,2), exact v^T M v=4/5

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Fem Quadratic Gauss QuadratureFem Mass Matrix Exactness

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010222Cubic Hermite cantilever L=2, EI=6, q=12 has tip (4,8/3), support residual (-24,-24), midpoint 4/3 not 17/12

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Fem Cubic Hermite InterpolationFem Consistent Beam Loading

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010323Two CCW triangles on the unit square give K with free u2=u4=1/2 and reactions r1=-1/2, r3=1/2

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Fem P1 Triangle StiffnessFem Global Scatter Reaction

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010324MPC u3=u1+u2 with K=diag(2,3,4) reduces to q=(7/13,9/13) and lambda=-12/13

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Fem Multipoint TransformationFem Kkt Multiplier Recovery

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010325Condensing internal dof 3 from the given 3x3 yields boundary (7/5,57/35) and internal 52/35

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Fem Static CondensationFem Interior Dof Recovery

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010426Qh=11/4,35/16,131/64 on h=1/2,1/4,1/8 has ratio 4, observed p=2, Richardson value 2, and C=3

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Fem Observed Convergence OrderFem Richardson Extrapolation

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Stability, consistency, a priori error, and adaptivityFinite Element Methods · Mixed review

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FEM:010427Dorfler theta=0.70 on squared indicators (16,4,1,1) marks only {K1} with share 8/11

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Fem Dorfler Bulk MarkingFem Squared Indicator Threshold

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Stability, consistency, a priori error, and adaptivityFinite Element Methods · Mixed review

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FEM:010528Linearized buckling K phi = P G phi has Pcr=3-sqrt(3) and G-normalized low mode

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Fem Linearized Buckling EigenvalueFem Geometric Stiffness Generalized Eigen

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Time-dependent and eigenvalue finite-element systemsFinite Element Methods · Mixed review

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FEM:010529One Newmark average-acceleration step on m=2,k=8 is u1=3/5 with energy 4

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Fem Newmark Average AccelerationFem Undamped Sdof Discrete Energy

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Time-dependent and eigenvalue finite-element systemsFinite Element Methods · Mixed review

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FEM:010630Two consistent-tangent Newton steps from u0=1/2 give u2=536/511, not the root 1

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Fem Consistent Tangent NewtonFem Residual Jacobian Recompute

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Nonlinear, constrained, and mixed formulationsFinite Element Methods · Mixed review

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FEM:010631Active-set contact on K[[2,-1],[-1,2]] is u=(0,3/2), lambda=(1/2,0), energy -9/4

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Fem Unilateral Contact Active SetFem Kkt Complementarity Contact

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Nonlinear, constrained, and mixed formulationsFinite Element Methods · Mixed review

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911011Let u(x) denote the axial displacement of a uniform elastic bar occupying 0<x<2.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911012Consider the boundary-value problem -\big((1+x)u'(x)\big)'=2 on 0<x<1, with u(0)=0 and (1+x)u'(1)=5.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911013A bar on 0\le x\le3 has piecewise-constant axial rigidity EA=2 for 0<x<1 and EA=5 for 1<x<3.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911014Solve -u''(x)=x on 0<x<1 with the inhomogeneous essential conditions u(0)=2 and u(1)=-1 by a Galerkin finite elemen…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911015An elastic bar of length 4 has constant axial rigidity EA=12 and is discretized by two linear elements with nodes x…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911016A single two-node linear bar element occupies 2\le x\le5, has cross-sectional area A=3, and has a spatially varying…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911017A stepped bar is represented by three nodes at x_1=0, x_2=2, and x_3=3.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911018Let a uniform bar on 0\le x\le1 have EA=1, distributed load f(x)=1+x, prescribed displacement u(0)=0, and a tractio…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911019The weak problem for a bar is: find u\in H^1(0,2) with u(0)=1 such that for every v\in H^1(0,2) satisfying v(0)=0.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911020A bar occupies 0\le x\le2, has constant EA=2, and is fixed at both ends: u(0)=u(2)=0.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911021A straight axial bar is modeled by three two-node linear bar elements connecting nodes at x = 0, 0.40, 1.00, and 1.…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911022Three collinear nodes have scalar horizontal displacements u1, u2, and u3.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911023A uniform axial bar of length 2.0 m and constant rigidity EA = 80 MN is fixed at x = 0 and represented by two equal…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911024A tapered axial member occupies 0 <= x <= 1.2 m, has Young's modulus E = 70 GPa, and has cross-sectional area A(x)…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911025A one-dimensional spring mesh has four nodes with positive displacement to the right.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911026A prismatic steel bar of length 900 mm, area 300 mm^2, and Young's modulus 200 GPa is divided into three equal two-…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911027Two axial bars form parallel load paths between a fixed node 1 and a common rigid connector represented by node 2.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911028A tapered bar of length 2.0 m has E = 100 GPa and area A(x) = 400(1 + 0.5x) mm^2, where x is in meters.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911029A uniform axial rod extends from x = 0 to x = 3.0 m, has EA = 45 MN, and is modeled by three equal two-node linear…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911030A stepped axial member is represented by four nodes at x = 0, 0.5, 1.0, and 1.8 m.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911031A prismatic Euler-Bernoulli beam element has length L=2.4\,\text{m} and local degree-of-freedom vector \mathbf d=[w…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911032A two-node Euler-Bernoulli beam element uses the Hermite interpolation and local degree ordering [w_1,\theta_1,w_2,…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911033A horizontal Euler-Bernoulli beam element of length L=3.6\,\text{m} has degrees of freedom [w_1,\theta_1,w_2,\theta…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911034An Euler-Bernoulli beam element has length L=3.0\,\text{m}, local degrees of freedom [w_1,\theta_1,w_2,\theta_2], a…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911035A single Euler-Bernoulli beam element spans 0\le x\le4.0\,\text{m} and has local degrees of freedom [w_1,\theta_1,w…

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Weak GalerkinFinite Element Methods Problem Solving

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Problem library

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FEM:010117Two P1 elements for -u''+u=1 with Robin: free A [[13/3,-23/12],[-23/12,25/6]], u=(0,63/109,114/109), reaction -148/109

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Fem Robin Reaction Diffusion AssemblyFem Constrained Row Discrete Reaction

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010118One-mode Ritz for -u''=x: a=1/4, uh=x(1-x)/4, min J=-1/96, energy-error squared 1/720

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Fem One Mode Ritz PoissonFem Energy Error Galerkin Orthogonality

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010119Symmetric Nitsche on one P1 interval: stiffness 3, load 6, uh=2(1-x), coercive here iff gamma>1

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Fem Symmetric Nitsche DirichletFem Nitsche One Mode Coercivity

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Mixed review

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FEM:010220P1 triangle (0,0),(2,0),(0,1) with k=3,q=6: K=[[15/4,-3/4,-3],[-3/4,3/4,0],[-3,0,3]], energy 39

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Fem P1 Triangle StiffnessFem Affine Gradient Energy

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010221Quadratic [0,1] Gauss: K exact at 2 pts, M2 singular on (2,-1,2), exact v^T M v=4/5

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Fem Quadratic Gauss QuadratureFem Mass Matrix Exactness

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010222Cubic Hermite cantilever L=2, EI=6, q=12 has tip (4,8/3), support residual (-24,-24), midpoint 4/3 not 17/12

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Fem Cubic Hermite InterpolationFem Consistent Beam Loading

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Reference elements, interpolation, and numerical quadratureFinite Element Methods · Mixed review

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FEM:010323Two CCW triangles on the unit square give K with free u2=u4=1/2 and reactions r1=-1/2, r3=1/2

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Fem P1 Triangle StiffnessFem Global Scatter Reaction

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010324MPC u3=u1+u2 with K=diag(2,3,4) reduces to q=(7/13,9/13) and lambda=-12/13

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Fem Multipoint TransformationFem Kkt Multiplier Recovery

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010325Condensing internal dof 3 from the given 3x3 yields boundary (7/5,57/35) and internal 52/35

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Fem Static CondensationFem Interior Dof Recovery

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Global assembly and essential or natural boundary conditionsFinite Element Methods · Mixed review

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FEM:010426Qh=11/4,35/16,131/64 on h=1/2,1/4,1/8 has ratio 4, observed p=2, Richardson value 2, and C=3

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Fem Observed Convergence OrderFem Richardson Extrapolation

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Stability, consistency, a priori error, and adaptivityFinite Element Methods · Mixed review

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FEM:010427Dorfler theta=0.70 on squared indicators (16,4,1,1) marks only {K1} with share 8/11

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Fem Dorfler Bulk MarkingFem Squared Indicator Threshold

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Stability, consistency, a priori error, and adaptivityFinite Element Methods · Mixed review

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FEM:010528Linearized buckling K phi = P G phi has Pcr=3-sqrt(3) and G-normalized low mode

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Fem Linearized Buckling EigenvalueFem Geometric Stiffness Generalized Eigen

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Time-dependent and eigenvalue finite-element systemsFinite Element Methods · Mixed review

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FEM:010529One Newmark average-acceleration step on m=2,k=8 is u1=3/5 with energy 4

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Fem Newmark Average AccelerationFem Undamped Sdof Discrete Energy

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Time-dependent and eigenvalue finite-element systemsFinite Element Methods · Mixed review

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FEM:010630Two consistent-tangent Newton steps from u0=1/2 give u2=536/511, not the root 1

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Fem Consistent Tangent NewtonFem Residual Jacobian Recompute

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Nonlinear, constrained, and mixed formulationsFinite Element Methods · Mixed review

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FEM:010631Active-set contact on K[[2,-1],[-1,2]] is u=(0,3/2), lambda=(1/2,0), energy -9/4

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Fem Unilateral Contact Active SetFem Kkt Complementarity Contact

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Nonlinear, constrained, and mixed formulationsFinite Element Methods · Mixed review

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911011Let u(x) denote the axial displacement of a uniform elastic bar occupying 0<x<2.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911012Consider the boundary-value problem -\big((1+x)u'(x)\big)'=2 on 0<x<1, with u(0)=0 and (1+x)u'(1)=5.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911013A bar on 0\le x\le3 has piecewise-constant axial rigidity EA=2 for 0<x<1 and EA=5 for 1<x<3.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911014Solve -u''(x)=x on 0<x<1 with the inhomogeneous essential conditions u(0)=2 and u(1)=-1 by a Galerkin finite elemen…

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911015An elastic bar of length 4 has constant axial rigidity EA=12 and is discretized by two linear elements with nodes x…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911016A single two-node linear bar element occupies 2\le x\le5, has cross-sectional area A=3, and has a spatially varying…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911017A stepped bar is represented by three nodes at x_1=0, x_2=2, and x_3=3.

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Weak GalerkinFinite Element Methods Problem Solving

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FEM:9911018Let a uniform bar on 0\le x\le1 have EA=1, distributed load f(x)=1+x, prescribed displacement u(0)=0, and a tractio…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911019The weak problem for a bar is: find u\in H^1(0,2) with u(0)=1 such that for every v\in H^1(0,2) satisfying v(0)=0.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911020A bar occupies 0\le x\le2, has constant EA=2, and is fixed at both ends: u(0)=u(2)=0.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911021A straight axial bar is modeled by three two-node linear bar elements connecting nodes at x = 0, 0.40, 1.00, and 1.…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911022Three collinear nodes have scalar horizontal displacements u1, u2, and u3.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911023A uniform axial bar of length 2.0 m and constant rigidity EA = 80 MN is fixed at x = 0 and represented by two equal…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911024A tapered axial member occupies 0 <= x <= 1.2 m, has Young's modulus E = 70 GPa, and has cross-sectional area A(x)…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911025A one-dimensional spring mesh has four nodes with positive displacement to the right.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911026A prismatic steel bar of length 900 mm, area 300 mm^2, and Young's modulus 200 GPa is divided into three equal two-…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911027Two axial bars form parallel load paths between a fixed node 1 and a common rigid connector represented by node 2.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911028A tapered bar of length 2.0 m has E = 100 GPa and area A(x) = 400(1 + 0.5x) mm^2, where x is in meters.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911029A uniform axial rod extends from x = 0 to x = 3.0 m, has EA = 45 MN, and is modeled by three equal two-node linear…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911030A stepped axial member is represented by four nodes at x = 0, 0.5, 1.0, and 1.8 m.

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911031A prismatic Euler-Bernoulli beam element has length L=2.4\,\text{m} and local degree-of-freedom vector \mathbf d=[w…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911032A two-node Euler-Bernoulli beam element uses the Hermite interpolation and local degree ordering [w_1,\theta_1,w_2,…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911033A horizontal Euler-Bernoulli beam element of length L=3.6\,\text{m} has degrees of freedom [w_1,\theta_1,w_2,\theta…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911034An Euler-Bernoulli beam element has length L=3.0\,\text{m}, local degrees of freedom [w_1,\theta_1,w_2,\theta_2], a…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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FEM:9911035A single Euler-Bernoulli beam element spans 0\le x\le4.0\,\text{m} and has local degrees of freedom [w_1,\theta_1,w…

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Weak GalerkinFinite Element Methods Problem Solving

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Weak formulations, Galerkin projection, and variational structureFinite Element Methods · Calculation

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Problem results

FEM:9911001Consider the boundary-value problem -u''(x)=x for 0<x<1, with u(0)=u(1)=0.

FEM:9911002For -u''(x)+u(x)=\sin(\pi x) on 0<x<1, subject to u(0)=u(1)=0, use the trial function u_h(x)=a\,x(1-x)+b\,x(1-x)(2x…

FEM:9911003Consider -u''(x)=1+x for 0<x<1, with u(0)=u(1)=0.

FEM:9911004Solve approximately the variable-coefficient problem -\frac{d}{dx}\!\left((1+x)u'(x)\right)=x for 0<x<1, with u(0)=…

FEM:9911005Consider -u''(x)+2u(x)=1 on 0<x<1, with nonhomogeneous boundary conditions u(0)=1 and u(1)=0.

FEM:9911006For the mixed-boundary problem -u''(x)+u(x)=x on 0<x<1, with u(0)=0 and u'(1)=1, use the Galerkin approximation u_h…

FEM:9911007Consider -u''(x)=x^2 for 0<x<1, with u(0)=u(1)=0.

FEM:9911008For -u''(x)+u(x)=1 on 0<x<1, with u(0)=u(1)=0, use u_h(x)=a\,x(1-x)+b\,x^2(1-x).

FEM:9911009Consider -u''(x)+u(x)=x on 0<x<1, with u(0)=u(1)=0, and restrict the approximation to u_h(x)=a\,x(1-x).

FEM:9911010Solve -u''(x)=1 on 0<x<1, with u(0)=u(1)=0, by the continuous piecewise-linear Galerkin finite element method on th…

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Problem results

FEM:9911001Consider the boundary-value problem -u''(x)=x for 0<x<1, with u(0)=u(1)=0.

FEM:9911002For -u''(x)+u(x)=\sin(\pi x) on 0<x<1, subject to u(0)=u(1)=0, use the trial function u_h(x)=a\,x(1-x)+b\,x(1-x)(2x…

FEM:9911003Consider -u''(x)=1+x for 0<x<1, with u(0)=u(1)=0.

FEM:9911004Solve approximately the variable-coefficient problem -\frac{d}{dx}\!\left((1+x)u'(x)\right)=x for 0<x<1, with u(0)=…

FEM:9911005Consider -u''(x)+2u(x)=1 on 0<x<1, with nonhomogeneous boundary conditions u(0)=1 and u(1)=0.

FEM:9911006For the mixed-boundary problem -u''(x)+u(x)=x on 0<x<1, with u(0)=0 and u'(1)=1, use the Galerkin approximation u_h…

FEM:9911007Consider -u''(x)=x^2 for 0<x<1, with u(0)=u(1)=0.

FEM:9911008For -u''(x)+u(x)=1 on 0<x<1, with u(0)=u(1)=0, use u_h(x)=a\,x(1-x)+b\,x^2(1-x).

FEM:9911009Consider -u''(x)+u(x)=x on 0<x<1, with u(0)=u(1)=0, and restrict the approximation to u_h(x)=a\,x(1-x).

FEM:9911010Solve -u''(x)=1 on 0<x<1, with u(0)=u(1)=0, by the continuous piecewise-linear Galerkin finite element method on th…